MATH 467 Partial Differential Equations Exercises - Millersville ...

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 at
http://banach.millersville.edu/˜bob/math467/mmm.pdf.
1. Find the general solution for the following first-order partial differential equation.
3ux +5uy −xyu = 0
2. Find the general solution for the following first-order partial differential equation.
ux −uy +yu = 0
3. Find the general solution for the following first-order partial differential equation.
ux +4uy −xu = x
4. Find the general solution for the following first-order partial differential equation.
−2ux +uy −yu = 0
5. Find the general solution for the following first-order partial differential equation.
xux −yuy +u = x
6. Find the general solution for the following first-order partial differential equation.
x 2 ux −2uy −xu = x 2
7. Find the general solution for the following first-order partial differential equation.
ux −xuy = 4
8. Find the general solution for the following first-order partial differential equation.
x 2 ux +xyuy +xu = x−y
9. Find the general solution for the following first-order partial differential equation.
ux +uy −u = y

10. Find the general solution for the following first-order partial differential equation.
ux −y 2 uy −yu = 0
11. Find the general solution for the following first-order partial differential equation.
ux +yuy +xu = 0
12. Find the general solution for the following first-order partial differential equation.
xux +yuy +2 = 0
13. For the following first-order linear partial differential equation find the general solution
and the solutions satisfying the side conditions.
(a) u(x,x) = x 2
(b) u(x,−x) = 1−x 2
3yux−2xuy = 0
(c) u(x,y) = 2x on the ellipse 2x 2 +3y 2 = 4
14. For the following first-order linear partial differential equation find the general solution
and the solutions satisfying the side conditions.
(a) u(x,−6x+2) = e x
(b) u(x,−x 2 ) = 1
(c) u(x,−6x) = −4x
ux −6uy = y
15. For the following first-order linear partial differential equation find the general solution
and the solutions satisfying the side conditions.
(a) u(x,3x) = cosx
(b) u(x,2x) = x
(c) u(x,x 2 ) = 1−x
4ux +8uy −u = 1
16. For the following first-order linear partial differential equation find the general solution
and the solutions satisfying the side conditions.
−4yux +uy −yu = 0

(a) u(x,y) = x 3 on the line x+2y = 3
(b) u(x,y) = −y on y 2 = x
(c) u(1−2y 2 ,y) = 2
17. For the following first-order linear partial differential equation find the general solution
and the solutions satisfying the side conditions.
yux+x 2 uy = xy
(a) u(x,y) = 4x on the curve y = (1/3)x 3/2
(b) u(x,y) = x 3 on curve 3y 2 = 2x 3
(c) u(x,0) = sinx
18. For the following first-order linear partial differential equation find the general solution
and the solutions satisfying the side conditions.
(a) u(x,4x) = x
(b) u(x,y) = −2y on curve y 3 = x 3 −2
(c) u(x,−x) = y 2
y 2 ux +x 2 uy = y 2
19. Show that if u(x,y) = x 2 +y 2 tan −1 (y/x) then
20. Determine the value of n so that
solves the PDE
xux +yuy −u = 0.
u(x,y) = x 3 tan −1
2 2 x −xy +y
x 2 +xy +y 2
xux +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
x 2 uxx +2xyuxy +y 2 uyy = 0.
22. Let F, G, and H be arbitrary differentiable functions and let u(x,y) = F(x − y) +
xG(x−y)+x 2 H(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 into
its 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 proportionality.
Show that if we make the change of variable u(x,t)−u0 = v(x,t)e αt where
α is a suitably chosen constant, the equation above can be transformed into the form
of 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 into
its 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 proportionality.
Suppose u0 = 0, the length of the bar is L = 1, the ends of the bar are
kept at temperature 0, and the initial temperature distribution is given by f(x) for
0 ≤ 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
(b) If f(x)g(y) = 0 verify that
f ′′ (x)g(y)+f ′ (x)g ′ (y)+f(x)g ′′ (y) = 0
− f′′ (x)
f(x) = g′ (y) f
g(y)
′ (x)
f(x) + g′′ (y)
g(y) .
(c) Show that if f′ (x)
f(x) is not constant, then g′ (y)
g(y)
(d) Show that g(y) = Ce λy and show that g′′ (y)
g(y) = λ2 .
is constant, say λ.
(e) Show that f ′′ (x)+λf ′ (x)+λ 2 f(x) = 0. Solve this ODE for f(x) and show that
u(x,y) =
Acos
λ √ 3
2 x
λ
+Bsin
√ 3
2 x
e λ(y−x/2)
26. A square plate of edge length a has its planar faces insulated. Three of its edges are
kept at temperature zero while the fourth is kept at constant temperature u0. Show
that the steady-state temperature distribution is given by
u(x,y) = 2u0
π
∞
k=1
(1−coskπ)sin(kπx/a)sinh(kπy/a)
ksinh(kπ)

27. A square plate of edge length a has its planar faces insulated. Three of its edges
are kept at temperature zero while the fourth is kept at temperature f(x). Find the
steady-state temperature distribution in the plate.
28. Find the Fourier Series for f(x) = x 2 on the interval [−L,L].
29. Use the result above to obtain the sums of the following series:
∞ 1
=
k2 k=1
∞ (−1) k+1
k2 =
k=1
∞ 1
=
(2k −1) 2
∞ 1
=
(2k) 2
k=1
30. Let f(x) = (x 2 −1) 2 for −1 ≤ x ≤ 1.
k=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 finite
series to within an error of 10 −4 ?
(c) Use the result above to find the sum of the following series.
31. Assuming that f(x) and f ′ (x) are defined on [−L,L], show that f ′ (x) is an even
function if f(x) is an odd function and f ′ (x) is an odd function if f(x) is an even
function.
32. Find all the real eigenvalues of the following boundary value problem.
∞
k=1
1
k 4
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 exists
a 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 positive
eigenvalues {λn} ∞ n=1 where
36. Show that if a /∈ Z that
for −π < x < π.
πcos(ax)
2asin(aπ)
37. For 0 < x < 2π show that
e x = e2π −1
π
38. Use the result above to show that
lim
n→∞ λn = 1
4 (2n−1)2 π 2 .
y ′′ +λy = 0 for 0 ≤ x ≤ 1
y(0) = 0
y(1) = y ′ (1)
1 cosx
= +
2a2 12 cos(2x)
−
−a2 22 cos(3x)
+
−a2 32 −···
−a2
1
2 +
π cosh(π −x)
·
2 sinhπ
∞
n=1
cos(nx)−nsin(nx)
n2
.
+1
= 1
2 +
∞
n=1
cos(nx)
n 2 +1 .
39. Use the result above to find the sum of the infinite series
∞ 1
n2 +1 .
n=1
40. Use the result above to find the sum of the infinite series
∞ 1
(n2 +1) 2.
n=1

41. Suppose u(x,t) solves utt = a 2 uxx with a = 0.
(a) Let α, β, x0, and t0 be constants, with α = 0. Show that the function v(x,t) =
u(αx+x0,βt+t0) satisfies
vtt = β2a2 vxx.
α2 (b) Foranyconstantw,let ˆx = cosh(w)x+asinh(w)tandˆt = a −1 sinh(w)x+cosh(w)t.
Show that x = cosh(w)ˆx−asinh(w)ˆt and t = −a −1 sinh(w)ˆx+cosh(w)ˆt.
(c) Define û(ˆx,ˆt) = u(x,t) and show that
utt −a 2 uxx = ûˆtˆt −a 2 ûˆxˆx.
42. Find all the product solutions of the boundary value problem below. Assume k > 0.
utt = a 2 uxx −kut for 0 ≤ x ≤ L, t ≥ 0
u(0,t) = 0
u(L,t) = 0
43. Consider the initial boundary value problem:
utt = a 2 uxx for 0 ≤ x ≤ L, t ≥ 0
u(0,t) = 0
u(L,t) = 0
πx
4πx
u(x,0) = 3sin −sin
L L
ut(x,0) = 1
2 sin
2πx
.
L
Find the Fourier Series solution and the solution according to D’Alembert’s formula
and show that they are equal.
44. Solve the initial boundary value problem:
utt = a 2 uxx for 0 ≤ x ≤ π, t ≥ 0
ux(0,t) = 0
ux(π,t) = 0
u(x,0) = cos 2 x
ut(x,0) = sin 2 x.
45. A string is stretched tightly between x = 0 and x = L. At t = 0 it is struck at the
position x = b where 0 < b < L in such a way that the initial velocity ut is given by
v0 for |x−b| < ǫ
ut(x,0) = 2ǫ
0 for |x−b| ≥ ǫ.

Findthe solution to the wave equation for this initial condition. Discuss the case where
ǫ → 0 + .
46. Define new coordinates in the xy-plane by
ˆx = ax+by +f
ˆy = cx+dy +g
where a, b, c, d, f, and g are constants with ad−bc = 0. Define û(ˆx,ˆy) = u(x,y).
(a) Show that if u is C 2 , then
uxx +uyy = (a 2 +b 2 )ûˆxˆx +2(ac+bd)ûˆxˆy +(c 2 +d 2 )ûˆyˆy.
(b) Suppose that (ˆx,ˆy) are the new coordinates obtained by rotating the original axes
by some angle θ in the counterclockwise direction. Verify that a = cosθ, b = sinθ,
c = −sinθ, and d = cosθ. Show that in this case
47. Solve the boundary value problem
uxx +uyy = ûˆxˆx +ûˆyˆy.
uxx +uyy = 0 for 0 < x < π and 0 < y < π
u(x,0) = sinx
u(x,π) = sinx
u(0,y) = siny
u(π,y) = siny.
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 following
boundary value problem.
uxx +uyy = 0 for 0 < x < 1 and 0 < y < 1
u(x,0) = 3sin(πx)+x
u(x,1) = 3x−1
u(0,y) = sin(2πy)−y
u(1,y) = y +1.
49. Solve the boundary value problem
uxx +uyy = 0 for 0 < x < π and 0 < y < π
u(x,0) = 0
u(x,π) = x(π −x)
u(0,y) = 0
u(π,y) = 0.

50. Solve the boundary value problem
uxx +uyy = 0 for 0 < x < π and 0 < y < π
uy(x,0) = cosx−2cos 2 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 and
outer radius 2 subject to the boundary conditions:
52. Solve the boundary value problem
53. Solve the boundary value problem
u(1,θ) = 3+4cos(2θ)
u(2,θ) = 5sinθ.
uxx +uyy = 0 for x 2 +y 2 < 1
u(1,θ) = −1+8cos 2 θ
u(r,θ+2π) = u(r,θ).
uxx +uyy = 0 for 1 < x 2 +y 2 < 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 on
the two flat faces. The boundary of the plate is given a temperature distribution of
f(θ) = 10θ 2 where the central angle θ ranges from −π to π. What is the steady-state
temperature 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
that
sinz = sin(a)cosh(b)+icos(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
3πy
u(x,y,0) = cosπ(x+y)−cosπ(x−y)+sin(2πx)sin .
5

57. A solid cube of edge length 1 and with heat diffusivity k is initially at temperature
100 ◦ C. At time t = 0 the cube is placed in anenvironment whose constant temperature
is 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 temperature
100 ◦ C. Five faces of the cube are insulated. At time t = 0 the cube is placed in an
environment whose constant temperature is 0 ◦ C. Find the temperature at the center
of 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+4sin 2 xcos 2 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
61. Consider 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 the
values of u have been specified at the eight corners of the solid. Find a solution of the
form
u(x,y,z) = axyz +bxy +cyz +dxz +ex+fy +gz +h
to the PDE.
62. Consider the partial differential equation
uxx +uyy +uzz = 0

on the solid cube where 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and 0 ≤ z ≤ 1. Suppose u obeys the
following boundary conditions.
Find a solution of the form
to the boundary value problem.
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
u(x,y,z) = Ax 2 +By 2 +Cz 2 +Dx+Ey +Fz
63. Convert the function u(x,y,z) = 1/ x 2 +y 2 +z 2 to spherical coordinates and show
that ∆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
and initial condition
u(1,t) = 0
u(ρ,0) = sin3 (πρ)
.
ρ