OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of Education
Advanced General Certificate of Education
MEI STRUCTURED MATHEMATICS
4758
Differential Equations
Friday
27 JANUARY 2006
Afternoon
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
MEI Examination Formulae and Tables (MF2)
TIME
1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer
booklet.
•
Answer any three questions.
•
You are permitted to use a graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
•
The total number of marks for this paper is 72.
This question paper consists of 3 printed pages and 1 blank page.
HN/3
© OCR 2006 [R/102/2661]
Registered Charity 1066969
[Turn over
2
1
In an electric circuit, the current, I amps, at time t seconds is modelled by the differential equation
d2I
dI
6 kI 6e– t,
2
dt
dt
where k is a positive constant which depends on the capacitor in the circuit.
(i) In the case k 8, find the general solution.
[8]
(ii) In the case k 9, find the solution given that initially the current is 1.5 amps and
State the limiting value of the current as t tends to infinity.
dI
0.
dt
[12]
(iii) Show that, for all positive values of k, the complementary function for this differential
equation will tend to zero as t tends to infinity.
[4]
2
Three differential equations are to be solved.
dy
3y 1
dx
dy
x2 3xy cos x
dx
dy
x2 3x ( y 0.1y 2 ) cos x
dx
x2
(1)
(2)
(3)
(i) By separating the variables, or otherwise, solve equation (1) to find y in terms of x, subject to
the condition y 0 when x 1. Hence calculate y when x 2, giving your answer correct
to three significant figures.
[8]
(ii) Solve equation (2) to find y in terms of x, subject to the condition y 0 when x 1. Hence
calculate y when x 2, giving your answer correct to three significant figures.
[11]
Euler’s method is used to solve equation (3). The algorithm is given by
xr1 xr h,
yr1 yr hy r .
The algorithm starts from x 1, y 0 with h 0.1, and gives y 0.034 411 when x 1.8.
(iii) Carry out two more steps of the algorithm to find an approximation for the value of y when
x 2. How could you find this value with greater accuracy?
[5]
4758 January 2006
3
3
A rock of mass m kg is dropped from a height of 50 m above the sea. The rock falls under the action
of its weight mg N and a resistance force R N, given by R 0.001mv 2, where v m s–1 is the velocity
of the rock.
At time t seconds, the rock has fallen a distance x m.
(i) Show that Newton’s second law gives the equation
v
dv
g 0.001v 2,
dx
justifying the signs of the terms.
[3]
(ii) Solve this differential equation to find v in terms of x. Hence show that the rock hits the water
at a speed of 30.54 m s–1, correct to two decimal places.
[7]
When the rock is in the water, the resistance to motion is modelled by R 2mv. Assume that there
is no instantaneous change in the velocity of the rock as it hits the water, and that the only forces
on the rock are its weight and the resistance.
(iii) Formulate and solve a differential equation to find a relationship between v and x while the
rock is under water (you are not required to find v in terms of x). How deep must the water
be in order for the velocity of the rock to be reduced to 5 m s–1?
[9]
(iv) Use your differential equation from part (iii) to deduce the terminal velocity of the rock under
water. Sketch a graph of v against x for the entire motion of the rock.
[5]
4
The following simultaneous differential equations are to be solved.
dx
x 2y sin t
dt
dy
4x 3y cos t
dt
d 2x
dx
2 5x 3sin t cos t.
2
dt
dt
[5]
(ii) Find the general solution for x in terms of t.
[10]
(i) Show that
(iii) Hence obtain the corresponding general solution for y.
[5]
(iv) Obtain approximate expressions for x and y in terms of t, valid for large t. Hence show that,
for large t, x is approximately equal to y. Show that, for small t, this is not necessarily the case.
[4]
4758 January 2006
OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of Education
Advanced General Certificate of Education
MEI STRUCTURED MATHEMATICS
4758
Differential Equations
Thursday
15 JUNE 2006
Afternoon
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
MEI Examination Formulae and Tables (MF2)
TIME
1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer
booklet.
•
Answer any three questions.
•
There is an insert for use in Question 3.
•
You are permitted to use a graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
•
The acceleration due to gravity is denoted by g m s –2. Unless otherwise instructed, when a
numerical value is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
•
The total number of marks for this paper is 72.
This question paper consists of 3 printed pages, 1 blank page and an insert.
HN/3
© OCR 2006 [R/102/2661]
Registered Charity 1066969
[Turn over
2
1
The displacement x at time t of an oscillating system from a fixed point is given by
..
.
x 2l x 5x 0,
where l 0.
(i) For what value of l is the motion simple harmonic? State the general solution in this case.
[3]
(ii) Find the range of values of l for which the system is under-damped.
[3]
Consider the case l 1.
(iii) Find the general solution of the differential equation.
[3]
.
When t 0, x x0 and x 0, where x0 is a positive constant.
(iv) Find the particular solution.
[4]
(v) Find the least positive value of t for which x 0.
[3]
Now consider the case l 3 with the same initial conditions.
(vi) Find the particular solution and show that it is never zero for t 0.
2
[8]
The positive quantities x, y and z are related and vary with time t, where t 0. The value of x is
described by the differential equation
dx
2x t 1.
dt
When t 0, x 1.
(i) Solve the equation to find x in terms of t.
The quantity y is related to x by the differential equation 2x
[9]
dy
y. When t 0, y 4.
dx
(ii) Solve the equation to find y in terms of x. Hence express y in terms of t.
[5]
dz
2z 6x. When t 0, z 3.
dx
(iii) Solve this equation for z in terms of x. Calculate the values of x, y and z when t 1, giving
your answers correct to 3 significant figures.
[10]
The quantity z is related to x by the differential equation x
4758 June 2006
3
3
Answer parts (i) and (ii) on the insert provided.
Two spherical bodies, Alpha and Beta, each of radius 1000 km, are in deep space. The point A is
on the surface of Alpha, and the point B is on the surface of Beta. These points are the closest
points on the two bodies and the distance AB has the constant value of 8000 km.
A probe is fired from A at a speed of V0 km s–1 in an attempt to reach B, travelling in a straight line.
At time t seconds after firing, the displacement of the probe from A is x km, and the velocity of the
probe is v km s–1.
The equation of motion for the probe is
v
dv
1
1
.
2
( 9000 x )
( 1000 x ) 2
dx
This differential equation is to be investigated first by means of a tangent field, shown on the insert.
(i) Show that the direction indicators are parallel to the v-axis when v 0 ( x 4000 ) . Show
also that the direction indicators are parallel to the x-axis when x 4000 ( v 0 ) . Hence
complete the tangent field on the insert, excluding the point ( 4000, 0 ) .
[6]
(ii) Sketch the solution curve through ( 0, 0.025 ) and the solution curve through ( 0, 0.05 ) . Hence
state what happens to the probe when the speed of projection is
(A) 0.025 km s–1,
(B) 0.05 km s–1.
[6]
(iii) Solve the differential equation to find v 2 in terms of x and V0 .
[6]
(iv) Given that the probe reaches B, state the value of x at which v 2 is least. Hence find from your
solution in part (iii) the range of values of V0 for which the probe reaches B.
[6]
4
The simultaneous differential equations
dx
2x y 3
dt
dy
5x 4y 18
dt
are to be solved for t 0.
(i) Show that
d2x
dx
2 3x –6.
2
dt
dt
[6]
(ii) Find the general solution for x in terms of t. Hence obtain the corresponding general solution
for y.
[9]
(iii) Given that x 4, y 17 when t 0, find the particular solutions for x and y and sketch a
graph of each solution.
[9]
4758 June 2006
Candidate Name
Centre Number
Candidate
Number
OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of Education
Advanced General Certificate of Education
4758
MEI STRUCTURED MATHEMATICS
Differential Equations
INSERT
Thursday
15 JUNE 2006
Afternoon
1 hours 30 minutes
INSTRUCTIONS TO CANDIDATES
•
This insert should be used in Question 3.
•
Write your name, centre number and candidate number in the spaces provided at the top of this
page and attach it to your answer booklet.
This insert consists of 2 printed pages.
HN/3
© OCR 2006
Registered Charity 1066969
[Turn over
2
Insert for use with Question 3
v
0.06
0.04
0.02
0
0
2000
4000
–0.02
–0.04
–0.06
-
4758 Insert June 2006
6000
8000
x
4758/01
ADVANCED GCE UNIT
MATHEMATICS (MEI)
Differential Equations
THURSDAY 25 JANUARY 2007
Morning
Time: 1 hour 30 minutes
Additional materials:
Answer booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer booklet.
•
Answer any three questions.
•
You are permitted to use a graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
•
The acceleration due to gravity is denoted by g m s–2. Unless otherwise instructed, when a
numerical value is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
The total number of marks for this paper is 72.
ADVICE TO CANDIDATES
•
Read each question carefully and make sure you know what you have to do before starting your
answer.
•
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
This document consists of 4 printed pages.
HN/4
© OCR 2007 [R/102/2661]
OCR is an exempt Charity
[Turn over
2
1
The differential equation
d 2y dy
2y ek t
dt
dt 2
is to be solved for t 0.
Consider the case k 2.
(i) Find the general solution.
[9]
(ii) Find the particular solution subject to the conditions y 0 when t 0, and y tends to zero as
t tends to infinity. Show that this solution is zero only when t 0 and sketch a graph of the
solution.
[7]
Consider now the case k 1.
(iii) Find the general solution. Find also the particular solution subject to the same conditions as in
part (ii). [You may assume that tet Æ 0 as t Æ . ]
[8]
2
The differential equation
dy
2y cot 2x k,
dx
(*)
where k is a constant, is to be solved for 0 x 12 p .
(i) Show that
d
( ln sin x ) cot x.
dx
[1]
(ii) In the case k 0, solve the differential equation by separating the variables to find the general
solution for y in terms of x.
[6]
Now assume that k 0.
(iii) Solve the differential equation to show that the general solution is
y A cosec 2x 12 k cot 2x
where A is an arbitrary constant.
[7]
(iv) Find the particular solution subject to the condition y 0 when x 14 p . Sketch the graph of
the solution for 0 x 12 p, showing the behaviour of y as x tends to zero.
[4]
(v) By using the double angle formulae for sin 2x and cos 2x, or otherwise, show that there is a
solution to (*) for which y tends to a finite limit as x tends to zero. State the solution and its
limiting value.
[6]
© OCR 2007
4758/01 Jan 07
3
3
In an experiment, a ball-bearing of mass m kg falls through a liquid. The ball-bearing is released
from rest and t seconds later its displacement is x m and its velocity is v m s–1. The forces acting
on the ball-bearing are its weight and a resistance force R N. Three models for R are to be considered.
In the first model, R mk1v, where k1 is a positive constant.
(i) Show that
g
dv
g k1v. Hence show that v ( 1 ek1t ) .
dt
k1
(ii) Find an expression for x in terms of t.
[7]
[4]
In the second model, R mk 2v 2, where k 2 is a positive constant.
(iii) Show that
v
dv
1 and hence find v in terms of x.
g k2v 2 dx
[7]
3
In the third model, R = mk3v 2 , where k 3 is a positive constant. Euler’s method is used to solve the
resulting differential equation
3
dv
= g - k 3v 2 .
dt
.
The algorithm is given by tr1 tr h , vr1 vr hvr.
(iv) Given k3 1.225 and using a step length of 0.1, perform two iterations of the algorithm to
estimate v when t 0.2.
[5]
The terminal velocity of the ball-bearing is 4 m s–1.
(v) Verify the value of k3 given in part (iv).
[1]
[Question 4 is printed overleaf.]
© OCR 2007
4758/01 Jan 07
[Turn over
4
4
The following simultaneous differential equations are to be solved.
dx
= -3 x - y + 10,
dt
dy
= 2 x - y + 5.
dt
(i) Find the values of x and y for which
(ii) Show that
dy
dx
0.
dt
dt
d 2x
dx
4 5x 5.
2
dt
dt
[3]
[5]
(iii) Find the general solution for x in terms of t. Hence obtain the corresponding general solution
for y.
[10]
(iv) Given that x y 0 when t 0, find the particular solutions.
[3]
(v) Sketch the graph of the solution for x, making clear the behaviour of the curve initially and
for large values of t.
[3]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate
(UCLES), which is itself a department of the University of Cambridge.
© OCR 2007
4758/01 Jan 07
4758/01
ADVANCED GCE UNIT
MATHEMATICS (MEI)
Differential Equations
MONDAY 18 JUNE 2007
Morning
Time: 1 hour 30 minutes
Additional materials:
Answer booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer booklet.
•
Answer any three questions.
•
You are permitted to use a graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
•
The acceleration due to gravity is denoted by g m s –2. Unless otherwise instructed, when a
numerical value is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
The total number of marks for this paper is 72.
•
There is an insert for use in Question 3.
ADVICE TO CANDIDATES
•
Read each question carefully and make sure that you know what you have to do before starting
your answer.
•
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
This document consists of 4 printed pages and an insert.
HN/6
© OCR 2007 [R/102/2661]
OCR is an exempt Charity
[Turn over
2
1
An object is suspended from one end of a vertical spring in a resistive medium. The other end of
the spring is made to oscillate and the differential equation describing the motion of the object is
.
..
y 4y 29y 3 cos t,
where y is the displacement at time t of the object from its equilibrium position.
(i) Find the general solution.
[11]
.
(ii) Find the particular solution subject to the conditions y y 0 when t 0. What is the
amplitude of the motion for large values of t?
[8]
(iii) Find the displacement and velocity of the object when t 10 p.
[2]
At t 10 p, the upper end of the spring stops oscillating and the differential equation describing
the motion of the object is now
..
.
y 4y 29y 0.
(iv) Write down the general solution. Describe briefly the motion for t 10 p.
2
[3]
The differential equation
x
dy
2y 1 x n,
dx
where n is a positive constant, is to be solved for x 0.
First suppose that n 2.
(i) Find the general solution for y in terms of x.
[8]
(ii) Use your general solution to find the limit of y as x Æ 0. Show how the value of this limit can
dy
be deduced from the differential equation, provided that
tends to a finite limit as x Æ 0.
dx
[3]
(iii) Find the particular solution given that y 12 when x 1. Sketch a graph of the solution in
the case n 1.
[4]
Now consider the case n 2.
(iv) Find y in terms of x, given that y has the same value at x 1 as at x 2.
© OCR 2007
4758/01 June 07
[9]
3
3
There is an insert for use with part (iii) of this question.
Water is draining from a tank. The depth of water in the tank is initially 1 m, and after t minutes
the depth is y m.
The depth is first modelled by the differential equation
dy
= - k y (1 + 0.1cos 25 t),
dt
where k is a constant.
(i) Find y in terms of t and k.
[8]
(ii) If the depth of water is 0.5 m after 1 minute, show that k 0.586 correct to three significant
figures. Hence calculate the depth after 2 minutes.
[4]
An alternative model is proposed, giving the differential equation
dy
= - 0.586
dt
(
)
y + 0.1cos 25 t .
(*)
The insert shows a tangent field for this differential equation.
(iii) Sketch the solution curve starting at ( 0, 1 ) and hence estimate the time for the tank to empty.
[4]
.
.
Euler’s method is now used. The algorithm is given by tr1 tr h, yr1 yr hyr , where y is
given by (*).
(iv) Using a step length of 0.1, verify that this gives an estimate of y 0.935 54 when t 0.1 for
the solution through ( 0, 1 ) and calculate an estimate for y when t 0.2.
[6]
(v) Use (*) to show that when the depth of water is less than 1 cm the model predicts that
positive for some values of t.
dy
is
dt
[2]
[Question 4 is printed overleaf.]
© OCR 2007
4758/01 June 07
[Turn over
4
4
The following simultaneous differential equations are to be solved.
dx
= - 5 x + 4 y + e -2t ,
dt
dy
= - 9 x + 7 y + 3e -2t .
dt
(i) Show that
d 2x
dx
2 x 3e2t.
2
dt
dt
[5]
(ii) Find the general solution for x in terms of t.
[8]
(iii) Hence obtain the corresponding general solution for y, simplifying your answer.
[4]
(iv) Given that x y 0 when t 0, find the particular solutions. Find the values of
when t 0. Sketch graphs of the solutions.
dy
dx
and
dt
dt
[7]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate
(UCLES), which is itself a department of the University of Cambridge.
© OCR 2007
4758/01 June 07
4758/01
ADVANCED GCE UNIT
MATHEMATICS (MEI)
Differential Equations
INSERT
MONDAY 18 JUNE 2007
Morning
Time: 1 hour 30 minutes
Candidate
Name
Centre
Number
Candidate
Number
INSTRUCTIONS TO CANDIDATES
•
•
This insert should be used in Question 3.
Write your name, centre number and candidate number in the spaces provided and attach the page
to your answer booklet.
This insert consists of 2 printed pages.
HN/6
© OCR 2007 [R/102/2661]
OCR is an exempt Charity
[Turn over
2
3
(iii)
y
1.0
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 t
Spare copy
y
1.0
0.8
0.6
0.4
0.2
0
© OCR 2007
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 t
4758/01 Insert June 07
4758/01
ADVANCED GCE
MATHEMATICS (MEI)
Differential Equations
THURSDAY 24 JANUARY 2008
Morning
Time: 1 hour 30 minutes
Additional materials (enclosed):
None
Additional materials (required):
Answer Booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
•
Write your name in capital letters, your Centre Number and Candidate Number in the spaces
provided on the Answer Booklet.
•
Read each question carefully and make sure you know what you have to do before starting
your answer.
Answer any three questions.
•
•
•
•
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
The acceleration due to gravity is denoted by g m s−2 . Unless otherwise instructed, when a
numerical value is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 72.
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
This document consists of 4 printed pages.
© OCR 2008 [R/102/2661]
OCR is an exempt Charity
[Turn over
2
1
The differential equation
d2 y
dy
+ 2 + y = f(t) is to be solved for t ≥ 0 subject to the conditions that
2
dt
dt
dy
= 0 and y = 0 when t = 0.
dt
Firstly consider the case f(t) = 2.
(i) Find the solution for y in terms of t.
[10]
Now consider the case f(t) = e−t .
(ii) Explain briefly why a particular integral cannot be of the form ae−t or ate−t . Find a particular
integral and hence solve the differential equation, subject to the given conditions.
[8]
(iii) For t > 0, show that y > 0 and find the maximum value of y. Hence sketch the solution for t ≥ 0.
[You may assume that tk e−t → 0 as t → ∞ for any k.]
[6]
2
A raindrop falls from rest through mist. Its velocity, v m s−1 vertically downwards, at time t seconds
after it starts to fall is modelled by the differential equation
(1 + t)
dv
+ 3v = (1 + t)g − 3.
dt
(i) Solve the differential equation to show that v = 14 g(1 + t) − 1 + (1 − 14 g)(1 + t)−3 .
[10]
The model is refined and the term −3 is replaced by the term −2v, giving the differential equation
(1 + t)
dv
+ 3v = (1 + t)g − 2v.
dt
(ii) Find the solution subject to the same initial conditions as before.
[9]
(iii) For each model, describe what happens to the acceleration of the raindrop as t → ∞.
[5]
© OCR 2008
4758/01 Jan08
3
3
The population, P, of a species at time t years is to be modelled by a differential equation. The initial
population is 2000.
At first the model
dP
= 0.5P is used.
dt
(i) Find P in terms of t.
[3]
To take account of observed fluctuations, the model is refined to give
dP
= 0.5P + 170 sin 2t.
dt
(ii) State the complementary function for this differential equation. Find a particular integral and
hence state the general solution.
[8]
(iii) Find the solution subject to the given initial condition.
[2]
2
dP
= 0.5P + P 3 sin 2t. This is to be solved using Euler’s method.
dt
= tr + h, Pr+1 = Pr + hṖr .
The model is further refined to give
The algorithm is given by tr+1
(iv) Using a step length of 0.1 and the given initial conditions, perform two iterations of the algorithm
to estimate the population when t = 0.2.
[4]
The population is observed to tend to a non-zero finite limit as t → ∞, so a further model is proposed,
given by
1
2
dP
P
= 0.5P1 −
.
dt
12 000
(v) Without solving the differential equation,
4
(A) find the limiting value of P as t → ∞,
[3]
(B) find the value of P for which the rate of population growth is greatest.
[4]
The simultaneous differential equations
dx
= −3x + y + 9,
dt
dy
= −5x + y + 15,
dt
are to be solved for t ≥ 0.
d2 x
dx
+ 2 + 2x = 6.
dt
dt 2
[5]
(ii) Find the general solution for x.
[7]
(iii) Hence find the corresponding general solution for y.
[3]
(iv) Find the solutions subject to the conditions that x = y = 0 when t = 0.
[4]
(v) Sketch, on separate axes, graphs of the solutions for t ≥ 0.
[5]
(i) Show that
© OCR 2008
4758/01 Jan08
4758/01
ADVANCED GCE
MATHEMATICS (MEI)
Differential Equations
THURSDAY 12 JUNE 2008
Morning
Time: 1 hour 30 minutes
Additional materials (enclosed):
None
Additional materials (required):
Answer Booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
•
Write your name in capital letters, your Centre Number and Candidate Number in the spaces
provided on the Answer Booklet.
•
Read each question carefully and make sure you know what you have to do before starting
your answer.
Answer any three questions.
•
•
•
•
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
The acceleration due to gravity is denoted by g m s−2 . Unless otherwise instructed, when a
numerical value is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 72.
You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
This document consists of 4 printed pages.
© OCR 2008 [R/102/2661]
OCR is an exempt Charity
[Turn over
2
1
Fig. 1 shows a particle of mass 2 kg suspended from a light vertical spring. At time t seconds its
displacement is x m below its equilibrium level and its velocity is v m s−1 vertically downwards. The
forces on the particle are
•
its weight, 2g N
•
the tension in the spring, 8(x + 0.25g) N
•
the resistance to motion, 2kv N where k is a
positive constant.
equilibrium
level
xm
Fig. 1
(i) Use Newton’s second law to write down the equation of motion for the particle, justifying the
signs of the terms. Hence show that the displacement is described by the differential equation
d2 x
dx
+k
+ 4x = 0.
2
dt
dt
[4]
The particle is initially at rest with x = 0.1.
(ii) In the case k = 0, state the general solution of the differential equation. Find the solution, subject
to the given initial conditions.
[4]
(iii) In the case k = 2, find the solution of the differential equation, subject to the given initial
conditions. Sketch a graph of the solution for t ≥ 0.
[11]
(iv) Find the range of values of k for which the system is over-damped. Sketch a possible graph of
the solution in such a case.
[5]
2
The radioactive substance X decays into the substance Y, which in turn decays into Z. At time t hours
the masses, in grams, of X, Y and Z are denoted by x, y and respectively.
Initially there is 8 g of X and there is no Y or Z present.
The differential equation modelling the decay of X is
dx
= −2x.
dt
(i) Find x in terms of t.
[3]
The differential equation modelling the amount of Y is
dy
= 2x − y.
dt
(ii) Using your expression for x found in part (i), solve this equation to find y in terms of t.
[9]
(iii) Show that y > 0 for t > 0. Sketch a graph of y for t ≥ 0.
[5]
The differential equation modelling the amount of Z is
d
= y.
dt
2
(iv) Without solving this equation, show that x + y + = 8. Hence show that = 81 − e−t .
[5]
(v) Calculate the time required for 99% of the total mass to become substance Z.
[2]
© OCR 2008
4758/01 Jun08
3
3
dy
+ ky = t, where k is a constant, is to be solved for t ≥ 1, subject to the
dt
condition y = 0 when t = 1.
The differential equation t
(i) When k ≠ −1, find the solution for y in terms of t and k.
[10]
(ii) Sketch a graph of the solution for k = 2.
[2]
(iii) When k = −1, find the solution for y in terms of t.
[5]
dy
− sin y = t, subject to the condition y = 0 when t = 1. This
dt
is to be solved by Euler’s method. The algorithm is given by tr+1 = tr + h, yr+1 = yr + hẏr .
Now consider the differential equation t
(iv) Using a step length of 0.1, perform two iterations of the algorithm to estimate the value of y when
t = 1.2.
[4]
If the algorithm is carried out with a step length of 0.05, the estimate for y when t = 1.2 is y ≈ 0.2138.
(v) Explain with a reason which of these two estimates for y when t = 1.2 is likely to be more
accurate. Hence, or otherwise, explain whether these estimates are likely to be overestimates or
underestimates.
[3]
4
The simultaneous differential equations
dx
= 4x − 6y − 9 sin t,
dt
dy
= 3x − 5y − 7 sin t,
dt
are to be solved.
(i) Show that
d2 x dx
+
− 2x = −9 cos t − 3 sin t.
dt 2 dt
[6]
(ii) Find the general solution for x.
[9]
(iii) Hence find the corresponding general solution for y.
[3]
It is given that x is bounded as t → ∞.
(iv) Show that y is also bounded as t → ∞.
[2]
(v) Given also that y = 0 when t = 0, find the particular solutions for x and y. Write down the
expressions for x and y as t → ∞.
[4]
© OCR 2008
4758/01 Jun08
ADVANCED GCE
4758/01
MATHEMATICS (MEI)
Differential Equations
Candidates answer on the Answer Booklet
OCR Supplied Materials:
•
8 page Answer Booklet
•
Graph paper
•
MEI Examination Formulae and Tables (MF2)
Other Materials Required:
None
Wednesday 21 January 2009
Afternoon
Duration: 1 hour 30 minutes
*475801*
*
4
7
5
8
0
1
*
INSTRUCTIONS TO CANDIDATES
•
•
•
•
•
•
•
•
Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided
on the Answer Booklet.
Use black ink. Pencil may be used for graphs and diagrams only.
Read each question carefully and make sure that you know what you have to do before starting your answer.
Answer any three questions.
Do not write in the bar codes.
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
−2
The acceleration due to gravity is denoted by g m s . Unless otherwise instructed, when a numerical value is
needed, use g = 9.8.
INFORMATION FOR CANDIDATES
•
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicate that a correct method is being used.
The total number of marks for this paper is 72.
This document consists of 4 pages. Any blank pages are indicated.
© OCR 2009 [R/102/2661]
RP–8H16
OCR is an exempt Charity
Turn over
2
1
The differential equation
d3 y
d2 y dy
+
2
−
− 2y = 2
dx3
dx2 dx
is to be solved.
(i) Write down the auxiliary equation. Show that −2 is a root of this equation and find the other two
roots. Hence write down the complementary function.
[6]
(ii) Find the general solution.
[3]
When x = 0, y = 0 and when x = ln 2, y = 0. As x → ∞, y tends to a finite limit.
(iii) Show that y = −2e−2x + 3e−x − 1.
[6]
(iv) Show that y = 0 only when x = 0 or ln 2. Show also that the graph of y against x has only one
stationary point, and determine its coordinates.
[5]
(v) Sketch the graph of the solution for x ≥ 0.
2
[4]
The differential equation
dy
cos x + y sin x = x cos2 x
dx
is to be solved for | x | < 12 π subject to the condition that y = 1 when x = 0.
(i) Find the solution.
[10]
(ii) Sketch the solution curve.
[2]
Now consider the differential equation
dy
cos x + y sin x = x cos x sin x
dx
for | x | < 12 π , subject to the condition that y = 1 when x = 0.
(iii) Use Euler’s method with a step length of 0.1 to estimate y when x = 0.2. The algorithm is given
[6]
by xr+1 = xr + h, yr+1 = yr + hyr′ .
(iv) Use the integrating factor method and the numerical approximation
ã
0.2
0
x tan x dx ≈ 0.002 688
to estimate y when x = 0.2.
© OCR 2009
[6]
4758/01 Jan09
3
3
An oil drum of mass 60 kg is dropped from rest from a point A which is at a height of 10 m above a
lake. The oil drum is modelled as a particle that moves vertically. When it is x m below A, its speed
is v m s−1 . Before it enters the water, the forces acting on it are its weight and a resistance force of
magnitude 14 v2 N.
(i) Show that
v
dv
1
=
2
240g − v dx 240
and hence find v2 in terms of x.
[9]
(ii) Show that the speed of the oil drum as it reaches the water is 13.71 m s−1 , correct to two decimal
places.
[1]
After it enters the water, the forces acting on the oil drum are its weight, a resistance force of magnitude
60v N and a buoyancy force of 90g N vertically upwards.
Assume that the initial speed in the water is 13.71 m s−1 and that the oil drum moves vertically.
4
(iii) Show that t seconds after entering the water its speed is given by v = 18.61e−t − 4.9.
[8]
(iv) Calculate the greatest depth below the surface of the water that the oil drum reaches.
[6]
The simultaneous differential equations
dx
= −3x − y + 7
dt
dy
= 2x − y + 2
dt
are to be solved for t ≥ 0.
dx dy
=
= 0.
dt
dt
[2]
dx
d2 x
+ 4 + 5x = 5.
2
dt
dt
[5]
(i) Find the values of x and y for which
(ii) Show that
(iii) Find the general solution for x.
[6]
(iv) Find the corresponding general solution for y.
[3]
When t = 0, x = 4 and y = 0.
(v) Find the solutions for x and y.
[3]
(vi) Sketch the graphs of x against t and y against t, for t ≥ 0. Explain how your solution to part (i)
relates to your graphs.
[5]
© OCR 2009
4758/01 Jan09
ADVANCED GCE
4758/01
MATHEMATICS (MEI)
Differential Equations
Candidates answer on the Answer Booklet
OCR Supplied Materials:
•
8 page Answer Booklet
•
Graph paper
•
MEI Examination Formulae and Tables (MF2)
Other Materials Required:
None
Wednesday 20 May 2009
Afternoon
Duration: 1 hour 30 minutes
*475801*
*
4
7
5
8
0
1
*
INSTRUCTIONS TO CANDIDATES
•
•
•
•
•
•
•
•
Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided
on the Answer Booklet.
Use black ink. Pencil may be used for graphs and diagrams only.
Read each question carefully and make sure that you know what you have to do before starting your answer.
Answer any three questions.
Do not write in the bar codes.
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
−2
The acceleration due to gravity is denoted by g m s . Unless otherwise instructed, when a numerical value is
needed, use g = 9.8.
INFORMATION FOR CANDIDATES
•
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicate that a correct method is being used.
The total number of marks for this paper is 72.
This document consists of 4 pages. Any blank pages are indicated.
© OCR 2009 [R/102/2661]
RP–8L16
OCR is an exempt Charity
Turn over
2
1
A car travels over a rough surface. The vertical motion of the front suspension is modelled by the
differential equation
d2 y
+ 25y = 20 cos 5t,
dt 2
where y is the vertical displacement of the top of the suspension and t is time.
(i) Find the general solution.
Initially y = 1 and
[8]
dy
= 0.
dt
(ii) Find the solution subject to these conditions.
[4]
(iii) Sketch the solution curve for t ≥ 0.
[4]
A refined model of the motion of the suspension is given by
d2 y
dy
+ 2 + 25y = 20 cos 5t.
2
dt
dt
(iv) Verify that y = 2 sin 5t is a particular integral for this differential equation. Hence find the general
solution.
[6]
(v) Compare the behaviour of the suspension predicted by the two models.
2
[2]
The differential equation
x
dy
sin x
+ 3y =
dx
x
is to be solved for x > 0.
(i) Find the general solution for y in terms of x.
[9]
As x → 0, y tends to a finite limit.
(ii) Use the approximations sin x ≈ x − 16 x3 and cos x ≈ 1 − 12 x2 (both valid for small x) to find the
value of the arbitrary constant and the limiting value of y as x → 0. Hence state the particular
solution.
[6]
(iii) Show that, when y = 0, tan x = x.
[2]
An alternative method of investigating the behaviour of y for small x is to use the approximation
sin x ≈ x − 16 x3 in the differential equation, giving
x − 16 x3
dy
x
+ 3y =
.
dx
x
(iv) Solve this differential equation and, given that y tends to a finite limit as x → 0, show that the
value of the limit is the same as that found in part (ii).
[7]
© OCR 2009
4758/01 Jun09
3
3
(a) An electric circuit has an inductor and a resistor in series with an alternating power source.
The circuit is switched on and after t seconds the current is I amps. The current satisfies the
differential equation
dI
2 + 4I = 3 cos 2t.
dt
(i) Find the complementary function and a particular integral. Hence state the general solution
[8]
for I in terms of t.
Initially the current is zero.
(ii) Find the particular solution.
[2]
(iii) Calculate the amplitude of the current for large values of t. Sketch the solution curve for
[4]
large values of t.
(b) The displacement, y, of a particle at time t satisfies the differential equation
dy
= 2 − 2y + e−t .
dt
You are not required to solve this differential equation.
The particle initially has displacement zero. The displacement has only one stationary value,
which is where y = 98 . Also the velocity of the particle tends to zero as t → ∞.
(i) Without solving the differential equation, use it to find
(A) the gradient of the solution curve when t = 0;
[2]
(B) the value of t at the stationary value of y;
[3]
(C) the limit of y as t → ∞.
[2]
(ii) Hence sketch the solution curve for t ≥ 0, illustrating these results.
4
[3]
The simultaneous differential equations
dx
= 7x + 6y + 2e−3t
dt
dy
= −12x − 10y + 5 sin t
dt
are to be solved for t ≥ 0.
(i) Show that
d2 x
dx
+ 3 + 2x = 14e−3t + 30 sin t.
2
dt
dt
[5]
(ii) Show that this differential equation has a particular integral of the form x = ae−3t − 9 cos t + 3 sin t,
where a is a constant to be determined.
Hence find the general solution for x in terms of t.
(iii) Find the corresponding general solution for y.
[8]
[4]
(iv) Show that, for large values of t, x = y when tan t ≈ k, where k is a constant to be determined. [4]
(v) Find the ratio of the amplitudes of y and x for large values of t.
© OCR 2009
4758/01 Jun09
[3]
ADVANCED GCE
4758/01
MATHEMATICS (MEI)
Differential Equations
Candidates answer on the Answer Booklet
OCR Supplied Materials:
•
8 page Answer Booklet
•
Insert for Question 2 (inserted)
•
MEI Examination Formulae and Tables (MF2)
Other Materials Required:
None
Wednesday 27 January 2010
Afternoon
Duration: 1 hour 30 minutes
*475801*
*
4
7
5
8
0
1
*
INSTRUCTIONS TO CANDIDATES
•
•
•
•
•
•
•
•
•
Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided
on the Answer Booklet.
Use black ink. Pencil may be used for graphs and diagrams only.
Read each question carefully and make sure that you know what you have to do before starting your answer.
Answer any three questions.
Do not write in the bar codes.
There is an insert for use in Question 2.
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
The acceleration due to gravity is denoted by g m s−2 . Unless otherwise instructed, when a numerical value
is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
•
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicate that a correct method is being used.
The total number of marks for this paper is 72.
This document consists of 4 pages. Any blank pages are indicated.
© OCR 2010 [R/102/2661]
RP–9E16
OCR is an exempt Charity
Turn over
2
1
A particle is attached to a spring and suspended vertically from an oscillating platform. The vertical
displacement, y, of the particle from a fixed point at time t is modelled by the differential equation
d2 y
dy
+ 6 + 9y = 0.5 sin t.
2
dt
dt
(i) Find the general solution.
[9]
Initially the displacement and velocity are both zero.
(ii) Find the solution.
[5]
(iii) Describe the motion of the particle for large values of t.
[2]
(iv) Find approximate values of the velocity and displacement at t = 20π .
[3]
The motion of the platform is stopped at t = 20π and the differential equation modelling the subsequent
motion of the particle is
d2 y
dy
+ 6 + 9y = 0.
2
dt
dt
(v) Write down the general solution. Sketch the solution curve for t > 20π .
2
[5]
There is an insert for use with part (b)(i) of this question.
(a) The differential equation
dy
− y tan x = tan x
dx
is to be solved for | x | < 12 π .
(i) Find the general solution.
[8]
(ii) Find the equation of the solution curve that passes through the origin and sketch the curve.
[4]
(b) The differential equation
dy
− y2 tan x = tan x
dx
is to be solved approximately, first by using a tangent field and then by Euler’s method.
(i) On the insert is a tangent field for the differential equation. Sketch the solution curves
[4]
through the origin and through (0, 1).
Euler’s method is now used, starting at x = 0, y = 1. The algorithm is given by xr+1 = xr + h,
yr+1 = yr + hyr′ .
(ii) Carry out two steps with a step length of 0.1 to verify that the algorithm gives x = 0.2,
y ≈ 1.0201.
[5]
(iii) Explain why it would be inappropriate to extend this numerical solution as far as x = 1.6.
[2]
(iv) How could the accuracy of the estimate found in part (b)(ii) be improved?
© OCR 2010
4758/01 Jan10
[1]
3
3
Fig. 3 shows a small ball projected from a point O over
horizontal ground. The forces acting on the ball are its weight
and air resistance. Its initial horizontal component of velocity
is v1 and its subsequent horizontal velocity ẋ is modelled by
the differential equation
y
dẋ
= −kẋ,
dt
x
O
where k is a positive constant.
Fig. 3
The units of displacement are metres and the units of time are
seconds.
(i) Solve this differential equation to find ẋ in terms of t and hence show that the horizontal
v
[8]
displacement from O is given by x = 1 1 − e−kt .
k
The ball’s initial vertical component of velocity is v2 and its subsequent vertical velocity ẏ is modelled
by the differential equation
dẏ
= −kẏ − g.
dt
(ii) Solve this differential equation to find ẏ in terms of t and hence show that the vertical displacement
kv + g
g
from O is given by y = 2 2
1 − e−kt − t.
[10]
k
k
(iii) Eliminate t between the expressions for x and y to show that y = kv2 + g
g
kx
x + 2 ln1 − .
kv1
v1
k
[4]
(iv) In the case v1 = v2 = 10, k = 0.1, determine whether the ball will pass over a 5 m high wall at a
horizontal distance 8 m from O.
[2]
4
The simultaneous differential equations
dx
= −3x − 4y + 23,
dt
dy
= 2x + y − 7
dt
are to be solved.
d2 x
dx
+ 2 + 5x = 5.
2
dt
dt
[5]
(ii) Find the general solution for x.
[7]
(iii) Find the corresponding general solution for y.
[4]
(i) Show that
When t = 0, x = 8 and y = 0.
(iv) Find the particular solutions for x and y.
[4]
(v) Show that for sufficiently large t, y is always greater than x.
[4]
© OCR 2010
4758/01 Jan10
4
THERE ARE NO QUESTIONS PRINTED ON THIS PAGE.
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public
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If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department
of the University of Cambridge.
© OCR 2010
4758/01 Jan10
ADVANCED GCE
4758/01
MATHEMATICS (MEI)
Differential Equations
INSERT for Question 2
Wednesday 27 January 2010
Afternoon
Duration: 1 hour 30 minutes
*475801*
*
4
7
5
8
0
1
*
INSTRUCTIONS TO CANDIDATES
•
•
•
•
Write your name clearly in capital letters, your Centre Number and Candidate Number in the boxes above.
Use black ink. Pencil may be used for graphs and diagrams only.
This insert should be used to answer Question 2 part (b)(i).
Write your answers to Question 2 part (b)(i) in the spaces provided in this insert, and attach it to your Answer
Booklet.
INFORMATION FOR CANDIDATES
•
This document consists of 2 pages. Any blank pages are indicated.
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OCR is an exempt Charity
Turn over
2
2
(b)
(i)
y
2
1
–1
0
1
x
–1
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public
website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department
of the University of Cambridge.
© OCR 2010
4758/01 Ins Jan10
ADVANCED GCE
4758/01
MATHEMATICS (MEI)
Differential Equations
Monday 24 May 2010
Afternoon
Candidates answer on the Answer Booklet
OCR Supplied Materials:
•
8 page Answer Booklet
•
MEI Examination Formulae and Tables (MF2)
Other Materials Required:
•
Scientific or graphical calculator
Duration: 1 hour 30 minutes
*475801*
*
4
7
5
8
0
1
*
INSTRUCTIONS TO CANDIDATES
•
•
•
•
•
•
•
•
Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided
on the Answer Booklet.
Use black ink. Pencil may be used for graphs and diagrams only.
Read each question carefully and make sure that you know what you have to do before starting your answer.
Answer any three questions.
Do not write in the bar codes.
You are permitted to use a graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
The acceleration due to gravity is denoted by g m s−2 . Unless otherwise instructed, when a numerical value
is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
•
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicate that a correct method is being used.
The total number of marks for this paper is 72.
This document consists of 4 pages. Any blank pages are indicated.
© OCR 2010 [R/102/2661]
2R–9J05
OCR is an exempt Charity
Turn over
2
1
The equation of a curve in the x-y plane satisfies the differential equation
dy
d2 y
+4
+ 8y = 32x2 .
2
d
x
dx
(i) Find the general solution.
[10]
The curve has a minimum point at the origin.
(ii) Find the equation of the curve.
[4]
(iii) Describe how the curve behaves for large negative values of x.
[2]
(iv) Write down an approximate expression for y, valid for large positive values of x.
[1]
(v) Sketch the curve.
[3]
(vi) Use the differential equation to show that any stationary point below the x-axis must be a
minimum.
[4]
2
(a)
(i) Find the general solution of
dy
+ 2y = e−2t .
dt
[6]
(ii) Find the solution of
dß
+ 2ß = y,
dt
where y is the general solution found in part (i), subject to the conditions that ß = 1 and
dß
= 0 when t = 0.
[7]
dt
(b) The differential equation
dx
+ 2x = sin t
dt
is to be solved.
(i) Find the complementary function and a particular integral. Hence state the general solution.
[6]
(ii) Find the solution that satisfies the condition
dx
= 0 when t = 0.
dt
(iii) Find approximate bounds between which x varies for large positive values of t.
© OCR 2010
4758/01 Jun10
[3]
[2]
3
3
Water is leaking from a small hole near the base of a tank. The height of the surface of the water
above the hole is y m at time t minutes.
(i) Consider first a cylindrical tank. The height of the water is modelled by the differential equation
√
dy
= −k y,
dt
where k is a positive constant. The height of water is initially 1 m and after 2 minutes it is 0.81 m.
Find y in terms of t, stating the range of values of t for which the solution is valid. Sketch the
solution curve.
[10]
(ii) Now consider water leaking from a conical tank. The height of the water is modelled by the
differential equation
√
dy
π y2
= −0.4 y.
dt
Find how long it takes the height to decrease from 1 m to 0.81 m.
[5]
(iii) Now consider water leaking from a spherical tank. The height of the water is modelled by the
differential equation
√
dy
π (ay − y2 )
= −0.4 y,
dt
where a is the diameter of the sphere.
This equation is to be solved by Euler’s method. The algorithm is given by tr+1 = tr + h,
yr+1 = yr + hẏr . The diameter is 2 m and initially the height is 1 m.
Use a step length of 0.1 to estimate the height after 0.2 minutes.
[5]
(iv) For any tank, the velocity of the water leaving the hole is proportional to the square root of the
height of the surface of the water above the hole.
By considering the rate of change of the volume of water, derive the differential equation
√
dy
= −k y
dt
for the cylindrical tank in part (i).
[4]
[Question 4 is printed overleaf.]
© OCR 2010
4758/01 Jun10
Turn over
4
4
At time t, the quantities x and y are modelled by the simultaneous differential equations
dx
= 2x − 5y + 9e−2t ,
dt
dy
= x − 4y + 3e−2t .
dt
(i) Show that
dx
d2 x
+ 2 − 3x = 3e−2t .
2
dt
dt
[5]
(ii) Find the general solution for x.
[8]
(iii) Find the corresponding general solution for y.
[4]
Initially x = 0 and y = 2.
(iv) Find the particular solutions.
[4]
(v) Describe the behaviour of the solutions as t → ∞.
State, with reasons, whether this behaviour is different if the initial value of y is just less than 2,
[3]
and the initial value of x is still 0.
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4758/01 Jun10
ADVANCED GCE
4758/01
MATHEMATICS (MEI)
Differential Equations
Candidates answer on the answer booklet.
OCR supplied materials:
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Other materials required:
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Wednesday 26 January 2011
Afternoon
Duration: 1 hour 30 minutes
*475801*
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4
7
5
8
0
1
*
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•
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•
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•
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Write your name, centre number and candidate number in the spaces provided on the
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Answer any three questions.
Do not write in the bar codes.
You are permitted to use a scientific or graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
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INFORMATION FOR CANDIDATES
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The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail
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Turn over
2
1
(a) The displacement, x m, of a particle at time t seconds is given by the differential equation
dx
d2 x
+2
+ 5x = 4et .
2
d
t
dt
(i) Find the general solution.
[9]
The particle is initially at rest at the origin.
(ii) Find the particular solution.
[4]
(b) The differential equation
d2 y dy
d3 y
+
4
+
− 6y = 0
dx3
dx2 dx
is to be solved.
(i) Show that 1 is a root of the auxiliary equation and find the other two roots. Hence find the
general solution.
[5]
When x = 0, y = 1 and
2
dy
= −4. As x → ∞, y → 0.
dx
(ii) Find the particular solution subject to these conditions.
[4]
(iii) Find the value of x for which y = 0.
[2]
(a) The differential equation
2
dy
+ 2xy = e−x sin x
dx
is to be solved subject to the condition x = 0, y = 1.
(i) Find the particular solution for y in terms of x.
[9]
(ii) Show that y > 0 for all x and that y has a stationary point when x = 0. State the limiting
value of y as | x | → ∞. Hence draw a simple sketch graph of the solution, given that the
stationary point at x = 0 is a maximum.
[6]
(b) The differential equation
dy
+ 2xy = 1
dx
is to be solved numerically subject to the condition x = 0, y = 1.
(i) Use Euler’s method with a step length of 0.1 to estimate y when x = 0.2. The algorithm is
given by xr+1 = xr + h, yr+1 = yr + hy′r .
[4]
(ii) Use the integrating factor method and the approximation ã
when x = 0.2.
© OCR 2011
4758/01 Jan11
0.2
0
ex dx ≈ 0.2027 to estimate y
2
[5]
3
3
The differential equation
dy
+ ky = cos 3x,
dx
where k is a constant, is to be solved.
(i) Find the complementary function. Hence find the general solution for y in terms of x and k. [8]
(ii) Find the particular solution subject to the condition that
dy
= 1 when x = 0.
dx
[4]
Now consider the differential equation
dy
+ ky = 2e−kx .
dx
(iii) Find the general solution.
[6]
Now consider the differential equation
dy
d2 y
−2
= 4e2x .
2
dx
dx
(iv) Using your answer to part (iii), or otherwise, solve this differential equation subject to the
dy
conditions that y = 0 and
= 1 when x = 0.
[6]
dx
4
The populations of foxes, x, and rabbits, y, on an island at time t are modelled by the simultaneous
differential equations
dx
= 0.1x + 0.1y,
dt
dy
= −0.2x + 0.3y.
dt
(i) Show that
dx
d2 x
− 0.4
+ 0.05x = 0.
2
dt
dt
[5]
(ii) Find the general solution for x.
[4]
(iii) Find the corresponding general solution for y.
[4]
Initially there are x0 foxes and y0 rabbits.
(iv) Find the particular solutions.
[4]
(v) In the case y0 = 10x0 , find the time at which the model predicts the rabbits will die out. Determine
whether the model predicts the foxes die out before the rabbits.
[7]
© OCR 2011
4758/01 Jan11
4
THERE ARE NO QUESTIONS PRINTED ON THIS PAGE.
Copyright Information
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Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department
of the University of Cambridge.
© OCR 2011
4758/01 Jan11
ADVANCED GCE
4758/01
MATHEMATICS (MEI)
Differential Equations
Candidates answer on the answer booklet.
OCR supplied materials:
• 8 page answer booklet
(sent with general stationery)
• MEI Examination Formulae and Tables (MF2)
Other materials required:
• Scientific or graphical calculator
Wednesday 18 May 2011
Morning
Duration: 1 hour 30 minutes
*475801*
*
4
7
5
8
0
1
*
INSTRUCTIONS TO CANDIDATES
•
•
•
•
•
•
•
•
Write your name, centre number and candidate number in the spaces provided on the
answer booklet. Please write clearly and in capital letters.
Use black ink. Pencil may be used for graphs and diagrams only.
Read each question carefully. Make sure you know what you have to do before starting
your answer.
Answer any three questions.
Do not write in the bar codes.
You are permitted to use a scientific or graphical calculator in this paper.
Final answers should be given to a degree of accuracy appropriate to the context.
The acceleration due to gravity is denoted by g m s−2. Unless otherwise instructed, when
a numerical value is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
•
•
•
•
The number of marks is given in brackets [ ] at the end of each question or part question.
You are advised that an answer may receive no marks unless you show sufficient detail
of the working to indicate that a correct method is being used.
The total number of marks for this paper is 72.
This document consists of 4 pages. Any blank pages are indicated.
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RP–0I23
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Turn over
2
1
The differential equation
d2 y
dy
+4
+ 3y = 13 cos 2t
2
dt
dt
(∗)
is to be solved.
(i) Find the general solution.
[9]
(ii) Find the particular solution, given that when t = 0, y and
dy
are both zero.
dt
[6]
Now consider the differential equation
d2 ß
dß
d3 ß
+
4
+3
= −26 sin 2t.
dt
dt3
dt2
(iii) Show that the general solution may be expressed as ß = y + c where y is the general solution of
[2]
(∗) and c is a constant.
(iv) When t = 0, ß = 2,
2
dß
d2 ß
= 0 and 2 = 13. Use these conditions to find the particular solution. [7]
dt
dt
(a) A curve in the x-y plane satisfies the differential equation
dy 2y p
−
= x
dx
x
for x > 0.
(i) Find the general solution for y in terms of x.
[8]
The curve passes through (1, 0).
(ii) Find the equation of this curve.
[2]
(iii) Find the coordinates of the stationary point of this curve and find the values to which y and
dy
tend as x → 0. Sketch the curve.
[6]
dx
(b) The differential equation
dy p 2
= x + y2
dx
is to be solved approximately by using a tangent field.
(i) Describe the shape of the isocline for which
dy
= 1.
dx
[2]
dy
dy
dy
(ii) Sketch, on the same axes, the isoclines for the cases
= 1,
= 2,
= 3. Use these
dx
dx
dx
isoclines to draw a tangent field.
[3]
© OCR 2011
(iii) Sketch the solution curve through (0, 1).
[1]
(iv) Sketch the solution curve through the origin.
[2]
4758/01 Jun11
3
3
(a) A particle of mass 2 kg moves on a horizontal straight line containing the origin O. When its
displacement is x m from O, it is subject to a force of magnitude 2k2 x N directed towards O,
where k is a positive constant.
(i) Show that the velocity, v m s−1 , of the particle satisfies the differential equation
v
dv
= −k2 x.
dx
[3]
The particle is at rest when x = a, where a is a positive constant.
(ii) Solve the differential equation, subject to this condition. Hence show that, while the particle
moves in the negative direction,
p
dx
[6]
= −k a2 − x2 .
dt
Initially the particle is at x = a.
(iii) Use the standard integral
ä p
x
dx = arcsin + c
a
a −x
1
2
2
to find x in terms of t, k and a.
[5]
(b) At time t s, the angle, θ rad, that a pendulum makes with the vertical satisfies the differential
equation
dω
ω
= −9 sin θ
dθ
dθ
where ω =
.
dt
(i) Solve the differential equation for ω in terms of θ subject to the condition ω = 0 when
θ = 13 π . Hence show that, while θ is decreasing,
p
dθ
= −3 2 cos θ − 1.
dt
[6]
(ii) Starting from θ = 13 π when t = 0, use Euler’s method with a step length of 0.1 to estimate
θ when t = 0.1. The algorithm is given by tr+1 = tr + h, θr+1 = θr + hθ˙ r . State whether this
algorithm can usefully be continued, justifying your answer.
[4]
[Question 4 is printed overleaf.]
© OCR 2011
4758/01 Jun11
Turn over
4
4
The quantities x and y at time t are modelled by the simultaneous differential equations
dx
= −3x − 2y + 3t,
dt
dy
= 2x + y + t + 2.
dt
(i) Show that
dx
d2 x
+2
+ x = −5t − 1.
2
dt
dt
[5]
(ii) Find the general solution for x.
[8]
(iii) Find the corresponding general solution for y.
[4]
When t = 0, x = 9 and y = 0.
(iv) Find the particular solutions.
[4]
(v) Find approximate expressions for x and y in terms of t, valid for large positive values of t.
[3]
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department
of the University of Cambridge.
© OCR 2011
4758/01 Jun11
Wednesday 25 January 2012 – Afternoon
A2 GCE MATHEMATICS (MEI)
4758/01
Differential Equations
QUESTION PAPER
* 4 7 3 3 2 1 0 1 1 2 *
Candidates answer on the Printed Answer Book.
OCR supplied materials:
•
Printed Answer Book 4758/01
•
MEI Examination Formulae and Tables (MF2)
Duration: 1 hour 30 minutes
Other materials required:
•
Scientific or graphical calculator
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
•
The Question Paper will be found in the centre of the Printed Answer Book.
•
Write your name, centre number and candidate number in the spaces provided on the
Printed Answer Book. Please write clearly and in capital letters.
•
Write your answer to each question in the space provided in the Printed Answer
Book. Additional paper may be used if necessary but you must clearly show your
candidate number, centre number and question number(s).
•
Use black ink. HB pencil may be used for graphs and diagrams only.
•
Read each question carefully. Make sure you know what you have to do before starting
your answer.
•
Answer any three questions.
•
Do not write in the bar codes.
•
You are permitted to use a scientific or graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
•
The acceleration due to gravity is denoted by g m s–2. Unless otherwise instructed, when
a numerical value is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
This information is the same on the Printed Answer Book and the Question Paper.
•
The number of marks is given in brackets [ ] at the end of each question or part question
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•
You are advised that an answer may receive no marks unless you show sufficient detail
of the working to indicate that a correct method is being used.
•
The total number of marks for this paper is 72.
•
The Printed Answer Book consists of 16 pages. The Question Paper consists of 4 pages.
Any blank pages are indicated.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
•
Do not send this Question Paper for marking; it should be retained in the centre or
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Turn over
2
1
Fig. 1 shows a particle of mass 0.5 kg hanging from a light vertical spring. At time t seconds its displacement
is x m below its equilibrium level and its velocity is v m s−1 vertically downwards. The forces on the particle
are
•
•
•
its weight, 0.5g N,
the tension in the spring, 2.5(x + 0.2g) N,
equilibrium
level
the resistance to motion, kv N, where k is a positive constant.
xm
Fig. 1
(i) Use Newton’s second law to write down the equation of motion for the particle, justifying the signs of
the terms. Hence show that the displacement is described by the differential equation
d2x
dx
+ 2k
+ 5x = 0.
dt2
dt
[4]
The particle is initially at rest with x = 0.1.
(ii) Find the set of values of k for which the system is
(A) over-damped,
(B) under-damped,
(C) critically damped.
In each of the cases (A) and (B), sketch a possible displacement-time graph of the motion.
(iii) Sketch a displacement-time graph of the motion of the particle in the case k = 0.
[7]
[1]
A subsequent motion of the particle is modelled by the differential equation
d2x
dx
+ 5x = sin 4t.
2 +2
dt
dt
(iv) Find the particular solution subject to the conditions that the particle is initially at rest with x = 0. [12]
© OCR 2012
4758/01 Jan12
3
2
A population of bacteria grows from an initial size of 1000. After t hours the size of the population is P.
After 10 hours the size of the population is 4000.
At first the rate of growth is modelled as being proportional to the size of the population.
(i) Write down a differential equation modelling the population growth and solve it for P in terms of t.
[4]
To allow for constraints on the population growth, the model is revised to give
dP
= kP(5000 − P),
dt
where k is a constant.
(ii) Solve this differential equation to find t in terms of P, subject to the given conditions.
[9]
(iii) Find the time it takes for the population to reach 4900, giving your answer in hours, correct to two
decimal places.
[1]
The model is further refined to give
dP
= 10−15P α (5000 − P),
dt
where α is a constant, and it is observed that the maximum rate of growth occurs when P = 4000.
(iv) Show that α = 4.
[5]
Starting from t = 10, P = 4000, Euler’s method is used with a step length of 0.2 to solve this differential
equation. The algorithm is given by tr + 1 = tr + h, Pr + 1 = Pr + hṖr .
(v) Continue the algorithm for two steps to estimate the size of the population when t = 10.4.
3
[5]
Consider the differential equation
dy
− y = x.
dx
(i) Sketch the isoclines
dy
= m for m = 0, ±1, ±2. Hence draw a sketch of the tangent field.
dx
[3]
(ii) State which of the isoclines is an asymptote to any solution curve.
[1]
(iii) Sketch on your tangent field the solution curves through (2, 0) and (0, −2).
[3]
(iv) Use the integrating factor method to solve the differential equation for y in terms of x, subject to the
condition y = 3 when x = 0.
[7]
Now consider the differential equation
dy
− y = sin x.
dx
(v) Find the complementary function and a particular integral. Hence state the general solution.
[6]
(vi) Find the solution subject to the condition y = 3 when x = 0 and sketch the solution curve.
[4]
© OCR 2012
4758/01 Jan12
Turn over
4
4
The simultaneous differential equations
dx
= −x + 2y
dt
dy
= −x −4y + e−2t
dt
are to be solved.
(i) Eliminate y to obtain a second order differential equation for x in terms of t. Hence find the general
solution for x.
[14]
(ii) Find the corresponding general solution for y.
[3]
Initially x = 5 and y = 0.
(iii) Find the particular solutions.
(iv) Show that
y
x
−
1
as t
2
[4]
∞. Show also that there is no value of t for which
y
1
=− .
2
x
[3]
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2012
4758/01 Jan12
Wednesday 16 May 2012 – Morning
A2 GCE MATHEMATICS (MEI)
4758/01
Differential Equations
QUESTION PAPER
* 4 7 1 5 7 6 0 6 1 2 *
Candidates answer on the Printed Answer Book.
OCR supplied materials:
•
Printed Answer Book 4758/01
•
MEI Examination Formulae and Tables (MF2)
Duration: 1 hour 30 minutes
Other materials required:
•
Scientific or graphical calculator
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
•
The Question Paper will be found in the centre of the Printed Answer Book.
•
Write your name, centre number and candidate number in the spaces provided on the
Printed Answer Book. Please write clearly and in capital letters.
•
Write your answer to each question in the space provided in the Printed Answer
Book. Additional paper may be used if necessary but you must clearly show your
candidate number, centre number and question number(s).
•
Use black ink. HB pencil may be used for graphs and diagrams only.
•
Read each question carefully. Make sure you know what you have to do before starting
your answer.
•
Answer any three questions.
•
Do not write in the bar codes.
•
You are permitted to use a scientific or graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
•
The acceleration due to gravity is denoted by g m s–2. Unless otherwise instructed, when
a numerical value is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
This information is the same on the Printed Answer Book and the Question Paper.
•
The number of marks is given in brackets [ ] at the end of each question or part question
on the Question Paper.
•
You are advised that an answer may receive no marks unless you show sufficient detail
of the working to indicate that a correct method is being used.
•
The total number of marks for this paper is 72.
•
The Printed Answer Book consists of 16 pages. The Question Paper consists of 4 pages.
Any blank pages are indicated.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
•
Do not send this Question Paper for marking; it should be retained in the centre or
recycled. Please contact OCR Copyright should you wish to re-use this document.
© OCR 2012 [R/102/2661]
DC (CW) 50327/2
OCR is an exempt Charity
Turn over
2
1
Some differential equations of the form
d2y
dy
+ 9y = f(x)
2 +6
dx
dx
are to be solved.
First consider the case f(x) = x2.
(i) Find the general solution for y in terms of x.
(ii) Find the particular solution subject to the conditions y = 0,
[9]
dy
= 0 when x = 0.
dx
[5]
Now consider the case f(x) = e−3x.
(iii) Explain why neither ae−3x nor axe−3x will be a particular integral for the differential equation.
[1]
(iv) State an appropriate form for a particular integral and hence find the general solution.
[7]
(v) State with reasons whether it is possible to have particular solutions for which
(A) y is positive for all values of x,
(B) y is negative for all values of x.
2
[2]
A parachutist of mass m kg falls vertically from rest. After she has fallen x m, her speed is v m s−1. The forces
acting on her are her weight and a resistance force of magnitude mkv2 N, where k is a constant.
(i) Show that her motion is modelled by the differential equation
v
dv
= g − kv2
dx
g
and solve this to show that v2 = k (1 − e−2kx).
(ii) Given that her terminal speed is 55 m s−1, calculate k.
[8]
[1]
When her speed is 54 m s−1, she opens her parachute. The motion is now modelled by assuming that the
magnitude of the resistance force instantaneously changes to 0.1mgv N. The time from the parachute
opening is t seconds.
(iii) Formulate and solve a differential equation to find v in terms of t.
[8]
(iv) Calculate the time it takes for her speed to reduce to 12 m s−1.
[1]
(v) Calculate the distance she falls from the point at which she opens her parachute to the point at which
her speed is 12 m s−1.
[6]
© OCR 2012
4758/01 Jun12
3
3
The differential equation x
dy
3
− 2y = x sin x is to be solved.
dx
(i) Find the general solution for y in terms of x.
[8]
(ii) Find the particular solution subject to the condition y = 0 when x = π. Sketch the solution curve
for 0 x 4π.
[5]
Now consider the differential equation x
dy
2
− 2y = 0.
dx
(iii) Find the general solution for y in terms of x.
Now consider the differential equation x
[5]
dy
2
3
− 2y = x sin x.
dx
This is to be solved numerically using Euler’s method. The algorithm is given by
xr + 1 = xr + h, yr + 1 = yr + hy′r with (x0, y0) = (3.14, 0).
4
(iv) Use a step length of 0.01 to estimate y when x = 3.16.
[5]
(v) How could this estimate be improved?
[1]
The simultaneous differential equations
dx
= −2x − y + 6,
dt
dy
= x − 2y + 7,
dt
are to be solved.
(i) Eliminate y to obtain a second order differential equation for x in terms of t. Hence find the general
solution for x.
[12]
(ii) Find the corresponding general solution for y.
[3]
Initially x = 7 and y = 0.
(iii) Find the particular solutions.
As t
y
∞, x
[4]
k.
(iv) State the value of k and show that y = kx for infinitely many values of t.
© OCR 2012
4758/01 Jun12
[5]
Monday 28 January 2013 – Morning
A2 GCE MATHEMATICS (MEI)
4758/01
Differential Equations
QUESTION PAPER
* 4 7 3 4 1 0 0 1 1 3 *
Candidates answer on the Printed Answer Book.
OCR supplied materials:
•
Printed Answer Book 4758/01
•
MEI Examination Formulae and Tables (MF2)
Duration: 1 hour 30 minutes
Other materials required:
•
Scientific or graphical calculator
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
•
The Question Paper will be found in the centre of the Printed Answer Book.
•
Write your name, centre number and candidate number in the spaces provided on the
Printed Answer Book. Please write clearly and in capital letters.
•
Write your answer to each question in the space provided in the Printed Answer
Book. Additional paper may be used if necessary but you must clearly show your
candidate number, centre number and question number(s).
•
Use black ink. HB pencil may be used for graphs and diagrams only.
•
Read each question carefully. Make sure you know what you have to do before starting
your answer.
•
Answer any three questions.
•
Do not write in the bar codes.
•
You are permitted to use a scientific or graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
•
The acceleration due to gravity is denoted by g m s–2. Unless otherwise instructed, when
a numerical value is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
This information is the same on the Printed Answer Book and the Question Paper.
•
The number of marks is given in brackets [ ] at the end of each question or part question
on the Question Paper.
•
You are advised that an answer may receive no marks unless you show sufficient detail
of the working to indicate that a correct method is being used.
•
The total number of marks for this paper is 72.
•
The Printed Answer Book consists of 16 pages. The Question Paper consists of 4 pages.
Any blank pages are indicated.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
•
Do not send this Question Paper for marking; it should be retained in the centre or
recycled. Please contact OCR Copyright should you wish to re-use this document.
© OCR 2013 [R/102/2661]
DC (SLM) 63805/3
OCR is an exempt Charity
Turn over
2
1
The differential equation
d3y
d2y
dy
+
2
- 5 - 6y = sin x
3
2
dx
dx
dx
is to be solved.
(i) Show that 2 is a root of the auxiliary equation. Find the other two roots and hence find the general
solution of the differential equation.
[10]
When x = 0 , y = 1 and
dy
= 0 . Also, y is bounded as x " 3.
dx
(ii) Find the particular solution.
(iii) Write down an approximate solution for large positive values of x. Calculate the amplitude of this
[4]
approximate solution and sketch the solution curve for large positive x.
Suppose instead that a solution is required that is bounded as x " - 3 .
(iv) Determine whether there is a solution for which y = 1 and
2
A ball of mass m kg falls vertically from rest through a liquid. At time t s, the velocity of the ball is v m s–1
and the ball has fallen a distance x m. The forces on the ball are its weight and a total upwards force of R N.
A student investigates three models for R.
In the first model R = mkv , where k is a positive constant.
(i) Show that
[6]
dy
= 0 when x = 0 .
dx
dv
= 9.8 - kv and hence find v in terms of t and k.
dt
[4]
[7]
The terminal velocity of the ball is observed to be 7 m s–1.
(ii) Find k.
[1]
In the second model, R = 0.2mv 2 .
(iii) Find v in terms of t. Show that your solution is consistent with a terminal velocity of 7 m s–1.
In the third model, R = 0.529mv 2 . Euler’s method is to be used to solve for v numerically.
The algorithm is given by t r + 1 = t r + h , v r + 1 = v r + hvo r with _ t 0, v 0 i = _ 0, 0 i .
(iv) Show that
(v) Show that this model is consistent with a terminal velocity of approximately 7 m s–1.
[10]
3
© OCR 2013
3
dv
= 9.8 - 0.529v 2 and find v when t = 0.2 using Euler’s method with a step length of 0.1.
dt
[5]
4758/01 Jan13
[1]
3
3
(a) Solve the differential equation
dy
- y tan x = sin x
dx
(b) Consider the differential equations
to find y in terms of x subject to the condition y = 1 when x = 0 .
(c) The differential equation
dy
+ f (x) y = g (x) ,
dx
(1)
dy
+ f (x) y = 0 .
dx
(2)
[9]
Show that if y = p (x) satisfies (1) and y = c (x) satisfies (2), then y = p (x) + Ac (x) satisfies (1),
where A is an arbitrary constant.
[5]
2 x2 + 1
d y 2y
n
+ = 2e x d
x
dx x
(3)
(i) Verify that y = e x satisfies (3).
(ii) Find the general solution of
(iii) Use the result of part (b) to find a solution of (3) for which y = 1 when x = 1.
4
The simultaneous differential equations
are to be solved.
is to be solved.
2
[3]
dy 2y
+ = 0 , giving y in terms of x.
dx x
[4]
[3]
dx
1
3
=- x - y + t
dt
2
2
dy 3
1
= x - y + 2t
dt
2
2
(i) Eliminate y to obtain a second order differential equation for x in terms of t. Hence find the general
solution for x.
[13]
(ii) Find the corresponding general solution for y.
[4]
When t = 0 , x = 1 and y = 0 .
(iii) Find the particular solutions.
(iv) Show that in this case x + y tends to a finite limit as t " 3 and state its value. Determine whether
[4]
x + y is equal to this limit for any values of t.
© OCR 2013
[3]
4758/01 Jan13
4
THEREARENOQUESTIONSPRINTEDONTHISPAGE.
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2013
4758/01 Jan13
Friday 17 May 2013 – Morning
A2 GCE MATHEMATICS (MEI)
4758/01
Differential Equations
QUESTION PAPER
* 4 7 1 5 7 6 0 6 1 3 *
Candidates answer on the Printed Answer Book.
OCR supplied materials:
•
Printed Answer Book 4758/01
•
MEI Examination Formulae and Tables (MF2)
Duration: 1 hour 30 minutes
Other materials required:
•
Scientific or graphical calculator
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
•
The Question Paper will be found in the centre of the Printed Answer Book.
•
Write your name, centre number and candidate number in the spaces provided on the
Printed Answer Book. Please write clearly and in capital letters.
•
Write your answer to each question in the space provided in the Printed Answer
Book. Additional paper may be used if necessary but you must clearly show your
candidate number, centre number and question number(s).
•
Use black ink. HB pencil may be used for graphs and diagrams only.
•
Read each question carefully. Make sure you know what you have to do before starting
your answer.
•
Answer any three questions.
•
Do not write in the bar codes.
•
You are permitted to use a scientific or graphical calculator in this paper.
•
Final answers should be given to a degree of accuracy appropriate to the context.
•
The acceleration due to gravity is denoted by g m s–2. Unless otherwise instructed, when
a numerical value is needed, use g = 9.8.
INFORMATION FOR CANDIDATES
This information is the same on the Printed Answer Book and the Question Paper.
•
The number of marks is given in brackets [ ] at the end of each question or part question
on the Question Paper.
•
You are advised that an answer may receive no marks unless you show sufficient detail
of the working to indicate that a correct method is being used.
•
The total number of marks for this paper is 72.
•
The Printed Answer Book consists of 16 pages. The Question Paper consists of 4 pages.
Any blank pages are indicated.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
•
Do not send this Question Paper for marking; it should be retained in the centre or
recycled. Please contact OCR Copyright should you wish to re-use this document.
© OCR 2013 [R/102/2661]
DC (LEG) 66177/3
OCR is an exempt Charity
Turn over
2
1
A particle is attached to a spring and suspended vertically from a point P which is made to oscillate vertically.
The vertical displacement, x, of the particle from a fixed point at time t is modelled by the differential
equation
2
d2x
dx
+ 3 + x = cos t.
dt
dt 2
(i) Find the general solution of the differential equation.
[8]
Initially the displacement and velocity of the particle are both zero.
(ii) Find the particular solution and sketch its graph for large positive values of t.
[6]
(iii) Find approximate values of the displacement and velocity at t = 10r .
[3]
The point P stops oscillating at t = 10r and the subsequent motion of the particle is modelled by
2
d2x
dx
+ 3 + x = 0.
2
dt
dt
(iv) Determine the type of damping present.
[2]
(v) Using the values obtained in part (iii), find the particular solution for this motion.
[5]
2
Inthisquestiontake g = 10.
A rocket of mass 500 kg is launched from rest from the sea bed at a depth of 124 m. It travels vertically
upwards. After t s it has risen x m and its velocity is v m s–1.
In a simple model, for all stages of its motion, the mass of the rocket is constant and the only forces acting
on it are its weight, a driving force of 10 000 N and a resistance force.
When in the sea, the magnitude of the resistance force is modelled by kv N, where k is a constant.
(i) Write down and solve a differential equation to show that v =
(ii) Find x in terms of t and k.
kt
5000 _
1 - e - 500 i .
k
[8]
[3]
The time for the rocket to reach the surface of the sea is 5 s.
(iii) Verify that k . 2.5 is consistent with this information and hence estimate the speed of the rocket when
it reaches the surface.
[3]
After the rocket reaches the surface it travels vertically upwards through the air and the magnitude of the
resistance force is now modelled by 0.4v 2 N.
(iv) Show that v
(v) Solve this differential equation to find the particular solution for v in terms of x. Sketch a graph of this
solution, showing the asymptote.
[8]
© OCR 2013
dv
= 10 - 0.0008v 2 .
dx
[2]
4758/01 Jun13
3
dy
+ 2y = sin 2x is to be solved.
dx
3
(a) The differential equation
(b) The differential equation
(i) Use the integrating factor method to find the general solution for y in terms of x.
[5]
(ii) Find the particular solution subject to the condition y = 2 when x = 0 .
[2]
(c) The differential equation
4
The simultaneous differential equations
(i) Find the complementary function and a particular integral. Hence write down the general solution.
[7]
(ii) Find the particular solution subject to the condition y = 2 when x = 0 . Sketch the solution curve
[4]
for x H 0 .
dy
+ 2y = e - x is to be solved.
dx
dy
+ 2y = tan x is to be solved subject to the condition y = 2 when x = 0 .
dx
Use an integrating factor and the approximation
value of y when x = 1.
y01 e 2x tan x dx . 2.71862 to calculate an approximate
[6]
dx
= x - 2y - z
dt
dy
= x + 3y + z
dt
dz
=-z
dt
are to be solved. When t = 0 , x = 1, y = 0 and z = 2 .
(i) Use the third equation to find the particular solution for z in terms of t.
(ii) Using part (i) eliminate y and z to obtain a second order differential equation for x. Hence find the
general solution for x in terms of t.
[12]
(iii) Find the corresponding general solution for y.
[3]
(iv) Find the particular solutions for x and y.
[4]
(v) Show that x = y when 3 sin t = e - 3t . Deduce that x = y occurs infinitely often.
[3]
© OCR 2013
4758/01 Jun13
[2]
4
THEREARENOQUESTIONSPRINTEDONTHISPAGE.
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2013
4758/01 Jun13