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Millersville University

Department of Mathematics

MATH 467 Partial Differential Equations

January 23, 2012

The most up-to-date version of this collection of homework exercises can always be found athttp://banach.millersville.edu/ bob/math467/mmm.pdf.

1. Find the general solution for the following first-order partial differential equation.

3ux + 5uy − xyu = 0


ux − uy + yu = 0


ux + 4uy − xu = x


−2ux + uy − yu = 0


xux − yuy + u = x


x2ux − 2uy − xu = x2


ux − xuy = 4


x2ux + xyuy + xu = x− y


ux + uy − u = y

http://banach.millersville.edu/~bob/math467/mmm.pdf


ux − y2uy − yu = 0


ux + yuy + xu = 0


xux + yuy + 2 = 0

13. For the following first-order linear partial differential equation find the general solutionand the solutions satisfying the side conditions.

3yux − 2xuy = 0

(a) u(x, x) = x2

(b) u(x,−x) = 1− x2

(c) u(x, y) = 2x on the ellipse 2x2 + 3y2 = 4


ux − 6uy = y

(a) u(x,−6x+ 2) = ex

(b) u(x,−x2) = 1

(c) u(x,−6x) = −4x


4ux + 8uy − u = 1

(a) u(x, 3x) = cos x

(b) u(x, 2x) = x

(c) u(x, x2) = 1− x


−4yux + uy − yu = 0

(a) u(x, y) = x3 on the line x+ 2y = 3

(b) u(x, y) = −y on y2 = x

(c) u(1− 2y2, y) = 2


yux + x2uy = xy

(a) u(x, y) = 4x on the curve y = (1/3)x3/2

(b) u(x, y) = x3 on curve 3y2 = 2x3

(c) u(x, 0) = sin x


y2ux + x2uy = y2

(a) u(x, 4x) = x

(b) u(x, y) = −2y on curve y3 = x3 − 2

(c) u(x,−x) = y2

19. Show that if u(x, y) =√

x2 + y2 tan−1(y/x) then

xux + yuy − u = 0.

20. Determine the value of n so that

u(x, y) = x3 tan−1

(

x2 − xy + y2

x2 + xy + y2

)

solves the PDExux + yuy − nu = 0.

21. Let F and G be arbitrary differentiable functions and let u(x, y) = F (y/x)+xG(y/x).Show that u(x, y) solves the PDE

x2uxx + 2xyuxy + y2uyy = 0.

22. Let F , G, and H be arbitrary differentiable functions and let u(x, y) = F (x − y) +xG(x− y) + x2H(x− y). Show that u(x, y) solves the PDE

uxxx + 3uxxy + 3uxyy + uyyy = 0.

23. The length of a metal rod is not insulated, but instead radiation can take place intoits surroundings. In this case the heat equation takes on the form:

ut = κuxx − c(u− u0)

where u0 is the constant temperature of the surroundings and c is a constant of propor-tionality. Show that if we make the change of variable u(x, t)− u0 = v(x, t)eαt whereα is a suitably chosen constant, the equation above can be transformed into the formof the heat equation for a rod whose length is insulated.

24. The length of a metal rod is not insulated, but instead radiation can take place intoits surroundings. In this case the heat equation takes on the form:

ut = κuxx − c(u− u0)

where u0 is the constant temperature of the surroundings and c is a constant of pro-portionality. Suppose u0 = 0, the length of the bar is L = 1, the ends of the bar arekept at temperature 0, and the initial temperature distribution is given by f(x) for0 ≤ x ≤ 1. Find u(x, t).

25. Consider the partial differential equation

uxx + uxy + uyy = 0.

(a) Let u(x, y) = f(x)g(y) and use the method of separation of variables to deduce

f ′′(x)g(y) + f ′(x)g′(y) + f(x)g′′(y) = 0

(b) If f(x)g(y) 6= 0 verify that

−f ′′(x)

f(x)=

g′(y)

g(y)

f ′(x)

f(x)+

g′′(y)

g(y).

(c) Show that iff ′(x)

f(x)is not constant, then

g′(y)

g(y)is constant, say λ.

(d) Show that g(y) = Ceλy and show thatg′′(y)

g(y)= λ2.

(e) Show that f ′′(x) + λf ′(x) + λ2f(x) = 0. Solve this ODE for f(x) and show that

u(x, y) =

(

A cos

(

λ√3

2x

)

+B sin

(

λ√3

2x

))

eλ(y−x/2)

26. A square plate of edge length a has its planar faces insulated. Three of its edges arekept at temperature zero while the fourth is kept at constant temperature u0. Showthat the steady-state temperature distribution is given by

u(x, y) =2u0

π

∞∑

k=1

(1− cos kπ) sin(kπx/a) sinh(kπy/a)

k sinh(kπ)

27. A square plate of edge length a has its planar faces insulated. Three of its edgesare kept at temperature zero while the fourth is kept at temperature f(x). Find thesteady-state temperature distribution in the plate.

28. Find the Fourier Series for f(x) = x2 on the interval [−L, L].

29. Use the result above to obtain the sums of the following series:

∞∑

k=1

1

k2=

∞∑

k=1

(−1)k+1

k2=

∞∑

k=1

1

(2k − 1)2=

∞∑

k=1

1

(2k)2=

30. Let f(x) = (x2 − 1)2 for −1 ≤ x ≤ 1.

(a) Find the Fourier Series for f(x) on [−1, 1].

(b) What is the minimum number of terms necessary to approximate f(x) by a finiteseries to within an error of 10−4?

(c) Use the result above to find the sum of the following series.

∞∑

k=1

1

k4

31. Assuming that f(x) and f ′(x) are defined on [−L, L], show that f ′(x) is an evenfunction if f(x) is an odd function and f ′(x) is an odd function if f(x) is an evenfunction.

32. Find all the real eigenvalues of the following boundary value problem.

y′′ + λy = 0 for 0 ≤ x ≤ 1

y(0) = y(1)

y′(0) = −y′(1)

33. Find all the real eigenvalues of the following boundary value problem.

y′′ + λy = 0 for 0 ≤ x ≤ π

πy(0) = y(π)

πy′(0) = −y′(π)

34. For the boundary value problem below, find all the values of L for which there existsa solution.

y′′ + y = 0 for 0 ≤ x ≤ L

y(0) = 0

y(L) = 1

35. For the boundary value problem below, show that there are infinitely many positiveeigenvalues {λn}∞n=1 where

limn→∞

λn =1

4(2n− 1)2π2.

y′′ + λy = 0 for 0 ≤ x ≤ 1

y(0) = 0

y(1) = y′(1)

36. Show that if a /∈ Z that

π cos(ax)

2a sin(aπ)=

1

2a2+

cosx

12 − a2− cos(2x)

22 − a2+

cos(3x)

32 − a2− · · ·

for −π < x < π.

37. For 0 < x < 2π show that

ex =e2π − 1

π

(

1

2+

∞∑

n=1

cos(nx)− n sin(nx)

n2 + 1

)

.

38. Use the result above to show that

π

2· cosh(π − x)

sinh π=

1

2+

∞∑

n=1

cos(nx)

n2 + 1.

39. Use the result above to find the sum of the infinite series

∞∑

n=1

1

n2 + 1.

40. Use the result above to find the sum of the infinite series

∞∑

n=1

1

(n2 + 1)2.

41. Suppose u(x, t) solves utt = a2uxx with a 6= 0.

(a) Let α, β, x0, and t0 be constants, with α 6= 0. Show that the function v(x, t) =u(αx+ x0, βt+ t0) satisfies

vtt =β2a2

α2vxx.

(b) For any constant w, let x = cosh(w)x+a sinh(w)t and t = a−1 sinh(w)x+cosh(w)t.Show that x = cosh(w)x− a sinh(w)t and t = −a−1 sinh(w)x+ cosh(w)t.

(c) Define u(x, t) = u(x, t) and show that

utt − a2uxx = utt − a2uxx.

42. Find all the product solutions of the boundary value problem below. Assume k > 0.

utt = a2uxx − kut for 0 ≤ x ≤ L, t ≥ 0

u(0, t) = 0

u(L, t) = 0

43. Consider the initial boundary value problem:

utt = a2uxx for 0 ≤ x ≤ L, t ≥ 0

u(0, t) = 0

u(L, t) = 0

u(x, 0) = 3 sin(πx

L

)

− sin

(

4πx

L

)

ut(x, 0) =1

2sin

(

2πx

L

)

.

Find the Fourier Series solution and the solution according to D’Alembert’s formulaand show that they are equal.

44. Solve the initial boundary value problem:

utt = a2uxx for 0 ≤ x ≤ π, t ≥ 0

ux(0, t) = 0

ux(π, t) = 0

u(x, 0) = cos2 x

ut(x, 0) = sin2 x.

45. A string is stretched tightly between x = 0 and x = L. At t = 0 it is struck at theposition x = b where 0 < b < L in such a way that the initial velocity ut is given by

ut(x, 0) =

{

v02ǫ

for |x− b| < ǫ0 for |x− b| ≥ ǫ.

Find the solution to the wave equation for this initial condition. Discuss the case whereǫ → 0+.

46. Define new coordinates in the xy-plane by

x = ax+ by + f

y = cx+ dy + g

where a, b, c, d, f , and g are constants with ad− bc 6= 0. Define u(x, y) = u(x, y).

(a) Show that if u is C2, then

uxx + uyy = (a2 + b2)uxx + 2(ac+ bd)uxy + (c2 + d2)uyy.

(b) Suppose that (x, y) are the new coordinates obtained by rotating the original axesby some angle θ in the counterclockwise direction. Verify that a = cos θ, b = sin θ,c = − sin θ, and d = cos θ. Show that in this case

uxx + uyy = uxx + uyy.

47. Solve the boundary value problem

uxx + uyy = 0 for 0 < x < π and 0 < y < π

u(x, 0) = sin x

u(x, π) = sin x

u(0, y) = sin y

u(π, y) = sin y.

48. Find a function of the form U(x, y) = a + bx + cy + dxy such that U(0, 0) = 0,U(1, 0) = 1, U(0, 1) = −1, and U(1, 1) = 2. Use this function to solve the followingboundary value problem.

uxx + uyy = 0 for 0 < x < 1 and 0 < y < 1

u(x, 0) = 3 sin(πx) + x

u(x, 1) = 3x− 1

u(0, y) = sin(2πy)− y

u(1, y) = y + 1.



u(x, 0) = 0

u(x, π) = x(π − x)

u(0, y) = 0

u(π, y) = 0.



uy(x, 0) = cos x− 2 cos2 x+ 1

uy(x, π) = 0

ux(0, y) = 0

ux(π, y) = 0.

51. Find the steady-state temperature distribution for an annulus of inner radius 1 andouter radius 2 subject to the boundary conditions:

u(1, θ) = 3 + 4 cos(2θ)

u(2, θ) = 5 sin θ.


uxx + uyy = 0 for x2 + y2 < 1

u(1, θ) = −1 + 8 cos2 θ

u(r, θ + 2π) = u(r, θ).


uxx + uyy = 0 for 1 < x2 + y2 < 2

u(1, θ) = a

u(2, θ) = b

u(r, θ + 2π) = u(r, θ).

54. A flat heating plate is in the shape of a disk of radius 5. The plate is insulated onthe two flat faces. The boundary of the plate is given a temperature distribution off(θ) = 10θ2 where the central angle θ ranges from −π to π. What is the steady-statetemperature at the center of the plate?

55. Let z = a + ib be a complex number (a and b are real numbers and i =√−1). Show

thatsin z = sin(a) cosh(b) + i cos(a) sinh(b).

56. Solve the initial boundary value problem:

ut = 2(uxx + uyy) for 0 ≤ x ≤ 3 and 0 ≤ y ≤ 5

u(x, 0, t) = 0

u(x, 5, t) = 0

u(0, y, t) = 0

u(3, y, t) = 0

u(x, y, 0) = cosπ(x+ y)− cosπ(x− y) + sin(2πx) sin

(

3πy

5

)

.

57. A solid cube of edge length 1 and with heat diffusivity k is initially at temperature100◦C. At time t = 0 the cube is placed in an environment whose constant temperatureis 0◦C. Find the temperature at the center of the cube as a function of time.

58. A solid cube of edge length 1 and with heat diffusivity k is initially at temperature100◦C. Five faces of the cube are insulated. At time t = 0 the cube is placed in anenvironment whose constant temperature is 0◦C. Find the temperature at the centerof the cube as a function of time.

59. Solve the boundary value problem:

uxx + uyy + uzz = 0 for 0 < x < π, 0 < y < π, 0 < z < π

ux(0, y, z) = 0

ux(π, y, z) = 0

uy(x, 0, z) = 0

uy(x, π, z) = 0

uz(x, y, 0) = 0

uz(x, y, π) = −1 + 4 sin2 x cos2 y.

60. Let f(x, t), g(y, t), and h(z, t) solve the respective heat equations

ft = kfxx

gt = kgyy

ht = khzz.

Show that u(x, y, z, t) = f(x, t)g(y, t)h(z, t) solves the partial differential equation

ut = k(uxx + uyy + uzz).


uxx + uyy + uzz = 0

on the rectangular solid where 0 ≤ x ≤ L, 0 ≤ y ≤ M , and 0 ≤ z ≤ N . Suppose thevalues of u have been specified at the eight corners of the solid. Find a solution of theform

u(x, y, z) = axyz + bxy + cyz + dxz + ex+ fy + gz + h

to the PDE.


uxx + uyy + uzz = 0

on the solid cube where 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and 0 ≤ z ≤ 1. Suppose u obeys thefollowing boundary conditions.

ux(0, y, z) = a0

ux(1, y, z) = a1

uy(x, 0, z) = b0

uy(x, 1, z) = b1

uz(x, y, 0) = c0

uz(x, y, 1) = c1

Find a solution of the form

u(x, y, z) = Ax2 +By2 + Cz2 +Dx+ Ey + Fz

to the boundary value problem.

63. Convert the function u(x, y, z) = 1/√

x2 + y2 + z2 to spherical coordinates and showthat ∆u = 0.

64. Convert the function u(x, y, z) = xyz to spherical coordinates and show that ∆u = 0.

65. Solve the heat equation on the solid sphere of radius 1 with boundary condition

u(1, t) = 0

and initial condition

u(ρ, 0) =sin3(πρ)

ρ.

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