Engineering Math, 1st exam, 2011/03/28 (8:00-10:00pm)
total 220 points
1. Classify the following differential equations as to type (either ordinary or partial
differential equation), order, and lineality.
(5 points for each)
(a) (sin xy) y´´´+ 4x(y´)2 = 0
d3 y
d2 y
dy
(b) x4(dx3 ) + 4(dx2 ) + x(dx) + ex = 5y
2. Solve ODEs.
(20 points for each)
(a) yy´ + xy2 = x
(b) (y cos xy – 2x) dx + (x cos xy + 2y) dy = 0
(c) 2xyy´ = 3y2 + x2,
y(1) = 2
(d) y´ + y tan x = e-0.01x cos x,
y(0) = 0
dy
(e) dx = y (xy3 - 1)
dy
(f) xdx + (3x+1)y = e-3x
3
3
(g) (1 - x + y) dx + (1 - y + x) dy = 0
3. The differential equation governing the velocity v of a falling weight m subjected to air
resistance proportional to the instantaneous velocity is “ m(dv/dt) = mg – kv ”, where k is a
positive constant of proportionality.
(a) Solve the equation subject to the initial condition v(0) = v0 and determine the limiting
velocity of the weight (in other words, velocity at t→∞).
(25 points)
(b) If distance s is related to velocity ds/dt = v , find an explicit expression for s if it further
known that s(0) = s0 .
(15 points)
4. A tank contains 800 gal of water in which 200 lb of salt is dissolved. Two gallons of fresh
water runs in per minute, and 2 gal of the mixture in the tank, kept uniform by stirring, runs
out per minute. How much salt is left in the tank at any time ?
(30 points)
Engineering Math, 2nd exam, 2011/04/27 (7:00-9:00pm)
total 220 points
1. Classify the following ordinary differential equations as to order and linearity.
(60 points, only if all answers are correct. No partial points)
d3 y
dy 4
(a) x�dx3 � – �dx� + y = 0
d2 y
(b) t5y´´´´– t3y´´ + 6y = 0
dy 2
d2 R
(d) dx2 = �1 + �dx�
k
(e) dt2 = – R2
d2 u
du
(c) dr2 + dr + u = cos (r + u)
(f) (sin θ) y´´´– (cos θ) y´ = 2
2. Suppose y1 and y2 form a fundamental set of solutions of the associated homogeneous form of
y´´+ P(x)y´ + Q(x)y = f(x). That is,
y1´´ + P(x)y1´ + Q(x)y1 = 0
y2´´ + P(x)y2´ + Q(x)y2 = 0
Derive the particular solution.
What is the general solution?
(20 points)
3. Is the set of functions f1(x) = 2 + x and f2(x) = 2 + |x| linearly dependent or linearly independent on
the interval (–∞, ∞)?
If they are not linearly dependent on (–∞, ∞), give an interval on which f1
and f2 are linearly dependent.
4. Solve ODEs.
(20 points)
(15 points for each)
(a) y´´– 2y´ + 2y = ex tan x
d2 y
(b) 2�dx2 � = 3y2,
y(0) = 1, y´(0) = 1
(c) y´´´– 5y´´ + 6y´ = 8 + 2 sin x
(d) x2y´´ + 3xy´ = 0,
y(1) = 0, y´(1) = 4
(e) (D2 + 5)x – 2y = 0
– 2x + (D2 + 2)y = 0
dx
(f) dt = – 5x – y
dy
dt
= 4x – y
x(1) = 0, y(1) = 1
5. How would you use the Cauchy-Euler method to solve (x + 2)2 y´´ + (x+2)y´+ y = 0 ?
Carry out
your ideas and give a solution.
(30 points)
* Thanks for your effort !
** Check your exam sheets and scores on 04/29 (this coming Friday, after 1:00pm) at the department office.
*** Please download and bring the lecture notes for chapter 5. New lecture notes were uploaded. (old ones were deleted!)
Engineering Math, Final exam, 2011/06/09 (7:00-9:00pm)
total 120 points
1. Answer true or false.
(10 points for each)
(a) The general solution of x2y´´+ xy´ + (x2 – 1)y = 0
is
y = c1J1(x) + c2J-1(x).
(b) Since x = 0 is an irregular singular point of x3y´´– xy´ + y = 0, the differential equation
possesses no solution that is analytic at x = 0.
2. Solve the following differential equations using the power series method.
(10 points for each)
(a) y´´ + (sin x)y = 0
(b) x2y´´ + xy´ + (16x2 – 4)y = 0 ;
x>0
3. Solve the following differential equations.
(a) y´´ – 2y´ + y = t δ(t – 1) ,
y(0) = 0, y´(0) = 0
(b) y´´ + 4y = δ(t – π) – δ(t – 2π) ,
t
dy
(c) dt + 6 y(t) + 9∫0 y(𝜏𝜏)𝑑𝑑𝜏𝜏 = 1 ,
(d) y´´ + 4y = (sin t) u(t – 1) ,
(10 points for each)
y(0) = 1, y´(0) = 0
y(0) = 0
y(0) = 1, y´(0) = 0
dx
(e) dt = x – 2y
dy
dt
= 5x – y
x(0) = –1,
y(0) = 2
4. Find the Laplace transform of the given function
(10 points for each)
(a)
(b)
S2
5. Derive the inverse transformation of (S2 +1)2 by using convolution theorem.
* I do appreciate your effort on the Engineering-Math course.
Engineering-Math course.
(10 points)
Enjoy your vacation, and see you in the second
** Please pick up your final-exam sheets at my office (3공-514) on June/13-14 (9:30-11:50am), and let me know if there
are any problems. Since then, no correction will be made !
Engineering Math, 1st exam, 2012/03/27 (7:00-9:00pm)
total 120 points
1. Classify the following ordinary differential equations as to order and linearity.
(20 points for 5 correct answers; 10 points for 4 correct answers; NO points for others)
(a) (sin θ) y´´´– (cos θ) y´ = 2y
d3 y
d2 y
dy
(b) x4(dx3 ) + 4(dx2 ) + x(dx) + ey = 5x
d2 R
k
(c) dt2 = – R2
d2 y
dy 2
(d) dx2 = �1 + �dx�
d2 u
du
(e) dr2 + dr + u = cos (r + u)
2. Solve ODEs.
(10 points for each)
(a) (y cos xy – 2x) dx + (x cos xy + 2y) dy = 0
(b) y´ = y (xy3 – 1)
y
(c) xy´ = x tan(x) + y
dy
(d) (x2 + 4)dx = (2x – 8xy)
(e) x dx + (x2y + 4y) dy = 0
(f) 2xyy´ = 3y2 + x2,
y(1) = 2
3. A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds
of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well-mixed solution is
pumped out at the same rate. Find the concentration c(t) of the salt in the tank at time t = 100
mins.
(20 points)
4. Determine whether the given set of functions is linearly dependent or linearly independent
on the interval (–∞, ∞). If they are not linearly dependent on (–∞, ∞), give an interval on
which functions are linearly dependent.
(a) f1(x) = 2 + x ,
(b) f1(x) = x ,
f2(x) = 2 + |x|
f2(x) = x2 ,
f3(x) = 4x – 3x2
(10 points for each)
Engineering Math, 2nd exam, 2012/05/01 (7:00-9:00pm)
total 120 points
1. Solve ODEs. (Do NOT use the Laplace transform !)
(10 points for each)
(a) x2y´´ + 3xy´ = 0,
y(1) = 0, y´(1) = 4
(b) (D4 – 2D3 – 8D2)y = 16 cos 2x
(c) y´´´ = �1 + (y´´)2
(use the method of reduction of order !)
(d) x´ = – 5x – y
y´ = 4x – y
x(1) = 0, y(1) = 1
2. Suppose y1 and y2 form a fundamental set of solutions of the associated homogeneous form of
y´´+ P(x)y´ + Q(x)y = f(x). That is,
y1´´ + P(x)y1´ + Q(x)y1 = 0
y2´´ + P(x)y2´ + Q(x)y2 = 0
Derive the particular solution.
What is the general solution?
(15 points)
3. Tank T1 and T2 in the figure contain initially 100 gal of
water each. In T1 the water is pure, whereas 150 lb of salt
are dissolved in T2. Liquid is pumped through the system
as indicated, and the mixtures are kept uniform by stirring.
Find the amounts of salt x(t) in T1 and y(t) in T2.
(30 points)
4. A projectile shot from a gun has weight w = mg and velocity v tangent
to its path of motion. Ignore air resistance and all other forces acting
on the projectile except its weight. Determine a system of differential
equations that describes its path of motion. Use Newton’s second
law of motion in the x and y directions.
(15 points)
5. Use the Laplace transform to solve the given initial-value problems.
(a) y´´ + 9y = et ,
y(0) = 0,
y´(0) = 0
(b) 2y´´´ + 3y´´ – 3y´ – 2y = e-t ,
y(0) = 0,
y´(0) = 0,
y´´(0) = 1
(10 points for each)
Engineering Math, Final exam, 2012/06/08 (7:00-9:00pm)
total 110 points
1. Solve the following differential equations using the Laplace transform.
(10 points for each)
t
(a) y(t) – ∫0 y(𝜏𝜏) sin 2(𝑡𝑡 − 𝜏𝜏) 𝑑𝑑𝜏𝜏 = sin 2t
(b) y´´ + 4y = δ(t – π) – δ(t – 2π) ,
y(0) = 1, y´(0) = 0
(c) x´ = y´ + u(t – π)
y´ = x + u(t + π)
x(0) = 1, y(0) = 0
2. Find L { f (t) } and L { e2t f (t) }.
(10 points for each)
(a)
(b)
(c)
3. Solve the following differential equations using the power series method.
(a) x2y´´ + xy´ + (2x2 – 64)y = 0 ;
x>0
(b) x2y´´ + 2xy´ + α2x2y = 0 ; x > 0,
(c) (x – 1)y´´ – xy´ + y = 0,
y = x-1/2 u(x) (use the indicated change of variable !)
y(0) = – 2, y´(0) = 6
(10 points)
(20 points)
(20 points)
Engineering Math, 1st exam, 2013/04/03 (7:00-9:00pm)
total 110 points
1. Classify the following ordinary differential equations as to order and linearity.
(20 points for 5 correct answers; 10 points for 4 correct answers; NO points for others)
(a)
d 3u
+ 4u =
sin t.
dt 3
d3 y
d2 y
dy
(b) x4(dx3 ) + 4(dx2 ) + x(dx) + ey = 5x
(c) e − x y '+ (4sin x) y =(tan x) y ''− 4e − x
d2 y
2
dy 2
(d) dx2 = �1 + �dx�
d2 u
du
(e) dr2 + dr + u = cos (r + k)
2. Solve ODEs.
dy
(10 points for each)
(a) 3(1 + t2) dt = 2ty (y3 – 1)
3
3
(b) (1 – y + x) y´ + y = x – 1
dy
(c) (cos x)dx + (sin x)y = 1
(d) y dx + x (ln x – ln y – 1) dy = 0 , y(1) = e
(e) (x2 + y2 – 5) dx = (y + xy) dy , y(0) = 1
y
(f) xy´ = ( x )3 + y
(g) y´ – 3y = – 12y2 , y(0) = 2
3. Suppose that a large mixing tank initially holds 500 gallons of water in which 70 pounds of
salt have been dissolved. Another salt solution is pumped into the tank at a rate of 6 gal/min,
and when the solution is well stirred, it is pumped out at a slower rate of 5 gal/min. If the
concentration of the solution entering is 3 lb/gal, find the amount of salt in the tank after 100
minutes.
(20 points)
Engineering Math, 2nd exam, 2013/05/04 (10:00-12:00am)
total 110 points
1. Is the set of functions f1(x) = x2 and f2(x) = x |x| linearly dependent or linearly independent on the
interval (–∞, ∞)?
If they are not linearly dependent on (–∞, ∞), give an interval on which f1 and
f2 are linearly dependent.
(10 points)
2. Solve ODEs. (Do NOT use the Laplace transform !)
(a) x2y´´ + 3xy´ = 0,
(10 points for each)
y(1) = 0, y´(1) = 4
(b) (D4 – 2D3 – 8D2)y = 16 cos 2x
(c) y´´´ = �1 + (y´´)2
d2 y
(d) 2�dx2 � = 3y2,
(use the method of reduction of order !)
y(0) = 1, y´(0) = 1
3. Initially, tanks T1 and T2 respectively contain 100 gal and
500 gal of water. In T1 the water is pure, whereas 150 lb of salt
are dissolved in T2. Liquid is pumped through the system
as indicated, and the mixtures are kept uniform by stirring.
(a) Find the amounts of salt x(t) in T1 and y(t) in T2.
(10 points)
(b) Suppose that both tanks T1 and T2 initially contain 100 gal of water for each. If you build up new
tank T3 of the same size as the others and connected to T2 by two tubes with flow rates as between
T1 and T2, what system of ODEs will you get?
(10 points)
(c) Find a general solution of the system in (b).
(20 points)
4. Use the Laplace transform to solve the given initial-value problems.
(a) y´´ – 4y´ = 6e3t – 3e-t ,
(b) y´ – y = 2 cos 5t ,
y(0) = 1,
y(0) = 0
y´(0) = – 1
(10 points for each)
Engineering Math, Final exam, 2013/06/15 (3:00-5:00pm)
total 100 points
1. Solve the following differential equations using the Laplace transform.
(10 points for each)
t
(a) y´(t) = 1 – sin t – ∫0 y(𝜏𝜏)𝑑𝑑𝜏𝜏 ,
(b) y´´– 2y´ = 1 + δ(t – 2) ,
y(0) = 0
y(0) = 0, y´(0) = 1
(c) y´´ + 4y = (sin t) u(t – 2π) ,
y(0) = 1, y´(0) = 0
(d) y´´ + 3y´+ 2y = u(t – 1) + δ(t – 2) ,
y(0) = 0, y´(0) = 1
2. Find L { et f (t) } of the given function.
(10 points)
3. Solve the following differential equations using the power series method.
(a) 3
d
dx
[xy ′ ] + (9x –
(b) y´´ + (sin x)y = 0
12
x
)y=0;
(c) x2y´´ + (α2x2 – v2 + 0.25)y = 0 ;
(d) (x + 2)y´´ + 3y = 0,
(10 points for each)
x>0
x > 0, y = x1/2 u(x) (use the indicated change of variable!)
y(0) = 0, y´(0) = 1
4. Using the solutions y1(x) and y2(x) of Legendre’s equation and the appropriate choice of c0 and c1,
find (a) the Legendre polynomials P6(x) and (b) the differential equation for which P6(x) is a
particular solution.
(10 points)
** Thank you for your effort on Engineering Math this semester.
Enjoy your summer vacation!!
Engineering Math, 1st exam, 2014/04/04 (7:00-9:00pm)
total 100 points
1. Classify the following ordinary differential equations as to order and linearity.
(20 points for 6 correct answers; 10 points for 5 correct answers; NO points for others)
d3 y
d2 y
dy
d2 y 3
dy
(a) (dx3 ) + (dx2 ) + 2x(dx) + ex = 5x2
(b) �1 + �dx2 � = dx
d3 R
k
(c) dt3 = – R2
d2 𝑢𝑢
d𝑢𝑢
(d) d𝑟𝑟 2 + d𝑟𝑟 = cos (r + k) + 3u
d3 w
(e) dt3 + w = et (tan w)
(f) (sin θ) y´´´– y´ + 4 = 2 (cos θ)
2. Solve ODEs.
(a) (
3y2 −x2
y5
(10 points for each)
x
)y´ + 2y4 = 0 ,
dy
y(1) = 1
(b) (t + 1) dt – ln t = – y ,
y(1) = 10
(d) y´ + (tan x)y = cos2 x ,
y(0) = – 1
(c) (ln x – ln y – 1)xy´ + y = 0 ,
dy
1
(e) xdx – y2 = – y ,
y(1) = e
y(1) = 2
3. A small cannonball weighting 16 lb is shot vertically upward with an initial velocity v0 =
160 ft/s. Assume that air resistance is ignored. If the positive direction is upward, then a
d2 𝑠𝑠
model for the state of the cannonball is given by d𝑡𝑡 2 = −g , where g = 32 ft/s2 and s(t) is the
height of the cannonball from ground level. Find the velocity v(t) of the cannonball at any
time t, and the maximum height (ft) attained by the cannonball.
(10 points)
4. Determine whether the given set of functions is linearly dependent or linearly independent
on the interval (–∞, ∞), and explain why.
If they are not linearly dependent on (–∞, ∞), give
an interval on which f1 and f2 are linearly dependent.
(a) f1(x) = 2 + x2 ,
(b) f1(x) = ex ,
f2(x) = 2 + x |x|
f2(x) = e-x ,
f3(x) = sinh x
(10 points for each)
Engineering Math, 2nd exam, 2013/05/09 (7:00-9:00pm)
total 110 points
1. Solve ODEs. (Do NOT use the Laplace transform !)
(10 points for each)
d2 x
(a) dt2 + w2x = F0 sin wt , x(0) = x´(0) = 0
(b) y´´ + 3y´ + 2y = sin ex
(c) y´´´ = �1 + (y´´)2
(use the method of reduction of order !)
(d) (x – 4)2 y´´– 5(x – 4)y´+ 9y = 0
(use the Cauchy-Euler method !)
2. Consider the differential equation x2y´´– (x2 + 2x)y´ + (x + 2)y = x3 . Note that y1 = x is one of
the solutions of the associated homogeneous equation. Find the general solution of the DE on the
interval (0,∞).
(25 points)
3. Tank T1 initially contains 400 gal of water
in which 120 lb of salt are dissolved. Tank T2
initially contains 200 gal of pure water.
Liquid
is pumped through the system as indicated, and
the mixtures are kept uniform by stirring.
Find the amounts of salt y1(t) in T1 , and y2(t) in T2 .
4. Use the Laplace transform to solve the given initial-value problems.
(a) y´´ – 3y´ + 2y = 4t – 8 ,
1
(b) y´´ – 4 y = 0 ,
y(0) = 2,
y(0) = 12,
y´(0) = 7
y´(0) = 0
(25 points)
(10 points for each)
Engineering Math, Final exam, 2014/06/07 (3:00-5:00pm)
total 100 points
1. Solve the following differential equations.
(10 points for each)
(a) y´´ + 2y´ + 5y = 25t – 100δ(t – π) ,
y(0) = –2, y´(0) = 5
𝑡𝑡
(b) y(t) + 2et∫0 y(𝜏𝜏)𝑒𝑒 −𝜏𝜏 𝑑𝑑𝜏𝜏 = tet
(c) y´´ – 5y´ + 6y = 6u(t – 1) ,
y(0) = 0, y´(0) = 0
(d) (cos x)y´´ + y = 0
d
3
x
(e) 3 dx [xy ′ ] – ( x − 3 )y = 0 ;
x>0
(f) 2xy´´+ y´ + y = 0
2. Find the Laplace transform of the given periodic function.
𝑒𝑒 −𝑎𝑎s
3. Derive the inverse transformation of S(S−2) by using convolution theorem.
(10 points)
(10 points)
4. Find (a) the Legendre polynomial P4(x) of (a2 – x2)y´´– 2xy´ + n(n+1)y = 0 (a ≠ 0) and (b) the
differential equation for which P4(x) is a particular solution. Please show how to convert the given
equation into Legendre equation by substitution, and obtain its solutions y1(x) and y2(x).
(20 points)
Engineering Math, 1st exam, 2015/04/09 (7:00-9:00pm)
total 110 points
1. Classify the following ordinary differential equations as to order and linearity.
(20 points for 6 correct answers; 10 points for 5 correct answers; NO points for others)
d3 w
(a) dt3 + w = et (tan w)
d2 y 3
dy
(b) �1 + �dx2 � = dx
d3 R
k
(c) dt3 = – R2
d2 𝑢𝑢
d𝑢𝑢
(d) d𝑟𝑟 2 + d𝑟𝑟 = cos (r + k) + 3u
d3 y
d2 y
dy
(e) (dx3 ) + (dx2 ) + 2x(dx) + ex = 5x2
(f) (sin θ) y´´´– y´ + 4 = 2 (cos θ)
2. Solve ODEs.
dy
(10 points for each)
(a) x 2 y −1 dx = 3y 3 + 2x ,
dy
1
y(1) = 2
(b) (6y + 4t – 1) dt + 4y = 5 – 2t ,
dT
(c) dt = k(T − Tm ) ,
T(0) = T0
y(–1) = 2
(K, Tm, and T0 are constants)
dy
(d) (cos x)dx + (sin x)y = 1
(e) x(ln x – ln y – 1) y´ = – y ,
y(1) = e
3. A differential equation describing the velocity (v) of a falling mass (m) subject to air
d𝑣𝑣
resistance proportional to the instantaneous velocity is m dt = mg − k𝑣𝑣 , where k > 0 is a
constant of proportionality called the drag force, and g is the acceleration due to gravity. The
positive direction is downward.
(10 points for each)
(a) Solve the equation subject to the initial condition v(0) = v0 .
(b) If the distance s, measured from the point where the mass was released above ground, is
related to velocity v by ds/dt = v, find an explicit expression for s(t) if s(0) = 0.
4. Determine whether the given set of functions is linearly dependent or linearly independent
on the interval (–∞, ∞), and explain why.
If they are not linearly dependent on (–∞, ∞), give
an interval on which f1 and f2 are linearly dependent.
(a) f1(x) = 2 + x2 ,
(b) f1(x) = x2 ,
f2(x) = 2 – x |x|
f2(x) = 1 – x2 ,
f3(x) = 2 + x2
(10 points for each)
Engineering Math, 2nd exam, 2015/05/04 (7:00-9:00pm)
total 120 points
1. Solve ODEs. (Do NOT use the Laplace transform !)
(10 points for each)
d2 x
(a) dt2 + w2x = F0 sin αt , x(0) = x´(0) = 0
(b) 4y´´ – y = xex/2 ,
y(0) = 1,
y´(0) = 0
(c) (x + 2)2 y´´+ (x + 2) y´ + y = 0
(use the substitution y = (x – x0)m)
2. Find a Cauchy-Euler differential equation ax2y´´ + bxy´ + cy = 0, where a, b, and c are real
constants, if it is known that…
(10 points, only if both are correct)
(a) m1 = 3 and m2 = – 1 are roots of its auxiliary equation.
(b) m1 = i is a complex root of its auxiliary equation.
3. Write down the general solution (y = yc + yp) of the differential equation, y´´ + w2y = sin αx, in
the two cases w ≠ α and w = α. Do not determine the coefficients in yp .
(10 points)
4. A mathematical model for the position x(t) of a body moving rectilinearly on the x-axis in an
d2 x
k2
inverse-square force field is given by dt2 = – x2 .
Suppose that at t = 0 the body starts from the
position x = x0, x0 > 0. What is the velocity of the body at time t ?
(20 points)
5. Initially, tanks T1 and T2 respectively contain 100 gal and
500 gal of water. In T1 the water is pure, whereas 150 lb of salt
are dissolved in T2. Liquid is pumped through the system
as indicated, and the mixtures are kept uniform by stirring.
(a) Find the amounts of salt x(t) in T1 and y(t) in T2.
(10 points)
(b) Suppose that both tanks T1 and T2 initially contain 100 gal of water for each. If you build up new
tank T3 of the same size as the others and connected to T2 by two tubes with flow rates as between
T1 and T2, what system of ODEs will you get?
(10 points)
(c) Find a general solution of the system in (b).
(10 points)
6. Use the Laplace transform to solve the given initial-value problems.
(a) y´´ + y = √2 sin √2t ,
(b) y´´ + 9y = et ,
y(0) = 0,
y(0) = 10,
y´(0) = 0
y´(0) = 0
(10 points for each)
Engineering Math, Final exam, 2015/06/09 (7:00-9:00pm)
total 110 points
1. Solve the following differential equations.
(10 points for each)
𝑡𝑡
1
(a) y(t) – ∫0 (t − 𝜏𝜏)y(𝜏𝜏)𝑑𝑑𝜏𝜏 = 2 − 2 𝑡𝑡 2
(b) y´´ + 4y´ + 5y = δ(t – 1) ,
y(0) = 0, y´(0) = 3
(c) y´´ + 4y = δ(t – π) – δ(t – 2π) ,
y(0) = 0, y´(0) = 1
(d) 2xy´´ + y´ + y = 0
(e) y´´ + ex y´ – y = 0
(f)
d
dx
4
[xy ′ ] + (2x – ) y = 0 ; x > 0
x
(g) 9x2y´´ + 9xy´ + (9x4 – 4)y = 0
(Hint: substitute
𝑥𝑥 2
2
)
2. Find the Laplace transform of the given function. Assume that a, b, and k are constants. (10 points)
k
3. Derive the inverse transformation of S2 (S2 −k2 ) by using convolution theorem.
(10 points)
4. Find (a) the Legendre polynomial P3(x) of (a2 – x2)y´´– 2xy´ + n(n+1)y = 0 (a ≠ 0) and (b) the
differential equation for which P3(x) is a particular solution. Please show how to convert the given
equation into Legendre equation by substitution, and obtain its solutions y1(x) and y2(x).
** Thank you for your effort on Engineering Math this semester.
(20 points)
Enjoy your summer vacation!!
Engineering Math, 1st exam, 2016/04/04 (7:30-9:30pm)
total 110 points
1. Classify the following ordinary differential equations as to order and linearity.
(20 points for 6 correct answers; 10 points for 5 correct answers; NO points for others)
d2 y 3
d3 w
d3 R
k
(c) dt3 = – R2
d3 y
d2 y
d2 𝑢𝑢
d𝑢𝑢
(d) d𝑟𝑟 2 + d𝑟𝑟 = cos (r + k) + 3u
dy
(f) (sin θ) y´´´– y´ + 4 = 2 (cos θ)
(e) (dx3 ) + (dx2 ) + 2x(dx) + ex = 5x2
2. Solve ODEs.
(10 points for each)
2𝑥𝑥+3𝑥𝑥 2 𝑦𝑦−𝑦𝑦 2 cos 𝑥𝑥
(a) y´= 2𝑦𝑦 sin 𝑥𝑥 −𝑥𝑥3 + ln 𝑦𝑦,
dT
(b) dt = k(T − Tm ) ,
y(0) = e
T(0) = T0
(c) x(ln x – ln y – 1) y´ = – y ,
dx
(d) (x 2 + 2𝑦𝑦 2 ) dy = xy ,
1
(e) y 2
dy
dy
(b) �1 + �dx2 � = dx
(a) dt3 + w = et (tan w)
3
+ y2 = 1 ,
dx
(k, Tm, and T0 are constants)
y(1) = e
y(–1) = 1
y(0) = 4
3. As a raindrop falls, it evaporates while retaining its spherical shape. If we make the further
assumptions that the rate at which the raindrop evaporates is proportional to its surface area
and that air resistance is negligible, then a model for the velocity v(t) of the raindrop is
d𝑣𝑣
d𝑡𝑡
+ 𝑘𝑘
𝑘𝑘
𝜌𝜌
3� �
� �𝑡𝑡 + 𝑟𝑟0
𝜌𝜌
𝑣𝑣 = g .
Here, 𝜌𝜌 is the density of water (raindrop), 𝑟𝑟0 is the radius of the
raindrop at t = 0, k < 0 is the constant of proportionality, and the downward direction is taken
to be the positive direction.
(10 points for each)
(a) Solve for v(t) if the raindrop falls from a cloud.
d𝑟𝑟
𝑘𝑘
(b) Assume that d𝑡𝑡 = 𝜌𝜌 .
If 𝑟𝑟0 = 0.01 ft and 𝑟𝑟 = 0.009 ft (10 seconds after the raindrop
falls from a cloud), determine the time at which the raindrop has evaporated completely.
4. Determine whether the given set of functions is linearly dependent or linearly independent
on the interval (–∞, ∞), and explain why.
If they are not linearly dependent on (–∞, ∞), give
an interval on which f1 and f2 are linearly dependent.
(a) f1(x) = 3 – x2 |x| ,
(b) f1(x) = cos 2x ,
f2(x) = 3 + x3
f2(x) = 1 ,
f3(x) = cos2 x
(10 points for each)
Engineering Math, 2nd exam, 2016/05/02 (7:30-9:30pm)
1. Solve ODEs. (Do NOT use the Laplace transform !)
d2 x
(a) dt2 + w2x = F0 cos αt ,
total 110 points
(10 points for each)
x(0) = x´(0) = 0
(b) y´´– 2y´ + y = 35x3/2ex
(c) (x2D2 + 11xD + 25)y = 0 ,
y(1) = 1, y´(1) = –1
(d) y´´ + 2y(y´)3 = 0
2. A mathematical model for the position x(t) of a body moving rectilinearly on the x-axis in an
d2 x
k2
inverse-square force field is given by dt2 = – x2 .
Suppose that at t = 0 the body starts from the
position x = x0, x0 > 0. What is the velocity of the body at time t ?
(10 points)
3. A very long cylindrical shell is formed by two concentric circular cylinders of different radii. A
chemically reactive fluid fills the space between the concentric cylinders as shown in black in the
figure. The inner cylinder has a radius of 1 and is thermally insulated, while the
outer cylinder has a radius of 2 and is maintained at a constant temperature T0 .
The rate of heat generation in the fluid due to the chemical reactions is
proportional to T/r2, where T(r) is the temperature of the fluid within the space
bounded between the cylinders defined by 1 < r < 2. Under these conditions, the
temperature of the fluid is defined by the following boundary-value problem:
1 𝑑𝑑
𝑑𝑑𝑑𝑑
𝑇𝑇
�𝑟𝑟 � = 2 , 1 < 𝑟𝑟 < 2,
𝑟𝑟 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
𝑟𝑟
𝑑𝑑𝑑𝑑
�
𝑑𝑑𝑑𝑑 𝑟𝑟=1
points)
= 0 , 𝑇𝑇(2) = 𝑇𝑇0 .
Find the temperature distribution T(r) within the fluid.
(20
4. Tank T1 initially contains 400 gal of water
in which 120 lb of salt are dissolved. Tank T2
initially contains 200 gal of pure water.
Liquid
is pumped through the system as indicated, and
the mixtures are kept uniform by stirring.
Find the amounts of salt y1(t) in T1 , and y2(t) in T2 .
(20 points)
5. Use the Laplace transform to solve the given initial-value problems.
(a) y´´ + y = √2 sin √2t ,
y(0) = 10,
(b) 2y´´´ + 3y´´ – 3y´ – 2y = e-t ,
y´(0) = 0
y(0) = 0,
y´(0) = 0,
y´´(0) = 1
(10 points for each)
Engineering Math, Final exam, 2016/06/06 (10:00-12:00am)
total 110 points
1. Solve the following differential equations.
(10 points for each)
𝑡𝑡
(a) y(t) + 2et∫0 y(𝜏𝜏)𝑒𝑒 −𝜏𝜏 𝑑𝑑𝜏𝜏 = tet
(b) y´´ + 4y´ + 5y = δ(t – 1) ,
y(0) = 0, y´(0) = 3
(c) (1 – x2)y´´– 2xy´ + 2y = 0
(d) 2xy´´– y´ + 2y = 0
(e) 4
d
dx
[xy ′ ] + (x –
16
x
)y=0;
(f) 9x2y´´ + 9xy´ + (9x4 – 4)y = 0
x>0
(Hint: substitute
𝑥𝑥 2
2
)
2. (a) Find L{et f (t)} of the given function.
(10 points)
t
(b) Find the Laplace transform of the staircase function in the Figure by noting that it is the
difference of at/b and the function given in the figure of (a).
(10 points)
f(t)
a
t
0
b
2b
3b
k
3. Derive the inverse transformation of S2 (S2 −k2 ) by using convolution theorem.
(10 points)
4. Find (a) the Legendre polynomial P5(x) of (a2 – x2)y´´– 2xy´ + n(n+1)y = 0 (a ≠ 0) and (b) the
differential equation for which P5(x) is a particular solution. Please show how to convert the given
equation into Legendre equation by substitution, and obtain its solutions y1(x) and y2(x). (20 points)