Differential Equations [132 marks]
1.
[Maximum mark: 21]
SPM.2.AHL.TZ0.11
A large tank initially contains pure water. Water containing salt begins to flow
into the tank The solution is kept uniform by stirring and leaves the tank through
an outlet at its base. Let x grams represent the amount of salt in the tank and let
t minutes represent the time since the salt water began flowing into the tank.
The rate of change of the amount of salt in the tank,
differential equation
(a)
(b)
(d)
(e)
dt
= 10e
−
4
−
x
t+1
dt
, is described by the
.
Show that t + 1 is an integrating factor for this differential
equation.
[2]
Hence, by solving this differential equation, show that
x (t) =
(c)
dx
t
dx
200−40e
−
t
4
t+1
(t+5)
.
[8]
Sketch the graph of x versus t for 0 ≤ t ≤ 60 and hence find the
maximum amount of salt in the tank and the value of t at which
this occurs.
[5]
Find the value of t at which the amount of salt in the tank is
decreasing most rapidly.
[2]
The rate of change of the amount of salt leaving the tank is
x
equal to t+1
.
Find the amount of salt that left the tank during the first 60
minutes.
[4]
2.
[Maximum mark: 18]
18N.3.AHL.TZ0.Hca_4
Consider the differential equation
(a)
dy
dx
= 1 +
y
x
, where x
≠ 0.
Given that y (1) = 1, use Euler’s method with step length h =
0.25 to find an approximation for y (2). Give your answer to
two significant figures.
[4]
(b)
Solve the equation
(c)
Find the percentage error when y (2) is approximated by the
final rounded value found in part (a). Give your answer to two
significant figures.
dy
dx
= 1 +
y
x
for y (1)
[6]
= 1.
[3]
Consider the family of curves which satisfy the differential equation
dy
dx
(d.i)
(d.ii)
= 1 +
y
x
, where x
≠ 0.
Find the equation of the isocline corresponding to
where k ≠ 0, k ∈ R.
dy
dx
= k,
Show that such an isocline can never be a normal to any of the
family of curves that satisfy the differential equation.
[1]
[4]
3.
[Maximum mark: 22]
Consider the differential equation
dy
dx
= f(
(a)
y
x
dv
f (v)−v
The curve y
dy
dx
), x > 0
Use the substitution y
∫
= vx to
= ln x + C
y +3xy+2x
x
2
2
.
− 1).
By using the result from part (a) or otherwise, solve the
differential equation and hence show that the curve has
equation y
(c)
= x(tan (ln x) − 1).
[9]
The curve has a point of inflexion at (x 1 ,
e
−
π
2
π
< x1 < e
2
y 1 ) where
. Determine the coordinates of this point of
inflexion.
(d)
[3]
gradient function given by
The curve passes through the point (1,
(b)
show that
where C is an arbitrary constant.
= f (x) for x > 0 has a
2
=
EXN.2.AHL.TZ0.12
Use the differential equation
[6]
dy
dx
2
=
y +3xy+2x
x
2
2
to show that
the points of zero gradient on the curve lie on two straight lines
of the form y = mx where the values of m are to be
determined.
[4]
4.
[Maximum mark: 35]
EXM.3.AHL.TZ0.4
This question investigates some applications of differential equations to
modeling population growth.
One model for population growth is to assume that the rate of change of the
population is proportional to the population, i.e.
the time (in years) and P is the population
(a)
dP
dt
= kP , where k ∈ R, t is
Show that the general solution of this differential equation is
P = Ae
kt
, where A
∈ R.
[5]
The initial population is 1000.
Given that k
= 0.003, use your answer from part (a)
to find
(b.i)
the population after 10 years
[2]
(b.ii)
the number of years it will take for the population to triple.
[2]
lim P
[1]
(b.iii)
t→∞
Consider now the situation when k is not a constant, but a function of time.
Given that k
(c.i)
= 0.003 + 0.002t, find
the solution of the differential equation, giving your answer in
the form P
(c.ii)
[5]
= f (t).
the number of years it will take for the population to triple.
Another model for population growth assumes
there is a maximum value for the population, L.
that k is not a constant, but is proportional to (1 −
P
L
).
[4]
(d)
Show that
(e)
Solve the differential equation
dP
dt
=
m
L
your answer in the form P
(f )
5.
[2]
P (L − P ), where m ∈ R.
dP
dt
=
= g (t).
m
L
P (L − P ), giving
[10]
Given that the initial population is 1000, L = 10000 and
m = 0.003, find the number of years it will take for the
population to triple.
[Maximum mark: 10]
[4]
22N.1.AHL.TZ0.9
Consider the homogeneous differential equation
dy
dx
2
=
y −2x
xy
2
, where
x, y ≠ 0.
It is given that y
(a)
= 2 when x = 1.
By using the substitution y
= vx, solve the differential
equation. Give your answer in the form y 2
The points of zero gradient on the curve y 2
the form y = mx where m ∈ R.
(b)
Find the values of m.
= f (x).
= f (x) lie on
[8]
two straight lines of
[2]
6.
7.
[Maximum mark: 7]
19M.3.AHL.TZ0.Hca_1
A simple model to predict the population of the world is set up as follows. At
time t years the population of the world is x, which can be assumed to be a
continuous variable. The rate of increase of x due to births is 0.056x and the rate
of decrease of x due to deaths is 0.035x.
(a)
Show that
(b)
Find a prediction for the number of years it will take for the
population of the world to double.
dx
dt
[1]
= 0.021x.
[6]
[Maximum mark: 19]
22M.2.AHL.TZ1.12
dy
Consider the differential equation x 2 dx = y 2
y > 2x. It is given that y = 3 when x = 1.
(a)
(b)
2
for x
> 0 and
Use Euler’s method, with a step length of 0. 1, to find an
approximate value of y when x = 1. 5.
Use the substitution y
x
(c.i)
− 2x
dv
dx
= v
2
= vx to
[4]
show that
[3]
− v − 2.
By solving the differential equation, show that y
(c.ii)
Find the actual value of y when x
(c.iii)
Using the graph of y
=
8x+x
4−x
4
3
=
8x+x
4−x
4
3
.
= 1. 5.
[10]
[1]
, suggest a reason why the
approximation given by Euler’s method in part (a) is not a good
estimate to the actual value of y at x = 1. 5.
© International Baccalaureate Organization, 2023
[1]