Download Homework 9

Problem 10.2.4 (p.613) Determine all solutions, if any, to the boundary value problem
y 00 + 9y = 0; 0 < x < π,
y(0) = 0, y 0 (π) = −6,
by first finding a general solution to the differential equation.
Problem 10.2.8 (p.613) Determine all solutions, if any, to the boundary value problem
y 00 − 2y 0 + y = 0; −1 < x < 1,
y(−1) = 0, y(1) = 2,
by first finding a general solution to the differential equation.
Problem 10.2.10 (p.613) Find the eigenvalues and the corresponding eigenfunctions for
the boundary value problem
y 00 + λy = 0; 0 < x < π,
y 0 (0) = 0, y(π) = 0.
Problem 10.2.12 (p.613) Find the eigenvalues and the corresponding eigenfunctions for
the boundary value problem
y 00 + λy = 0; 0 < x < π/2,
y 0 (0) = 0, y 0 (π/2) = 0.
Problem 10.2.14 (p.613) Find the eigenvalues and the corresponding eigenfunctions for
the boundary value problem
y 00 − 2y 0 + λy = 0; 0 < x < π,
y(0) = 0, y(π) = 0.
Problem 10.2.18 (p.613) Solve the heat flow problem
∂ 2u
∂u
(x, t) = 3 2 (x, t),
∂t
∂x
u(0, t) = u(π, t) = 0,
u(x, 0) = sin 4x + 3 sin 6x − sin 10x,
0 < x < π,
t > 0,
t > 0,
0 < x < π.
Problem 10.2.20 (p.614) Solve the vibrating string problem
2
∂ 2u
2∂ u
(x,
t)
=
3
(x, t),
∂t2
∂x2
u(0, t) = u(π, t) = 0,
u(x, 0) = 0,
∂u
(x, 0) = −2 sin 3x + 9 sin 7x − sin 10x,
∂t
0 < x < π,
t > 0,
0 < x < π,
0 < x < π.
t > 0,
Problem 10.2.24 (p.614) Find the formal solution to the vibrating string problem
2
∂ 2u
2∂ u
(x,
t)
=
4
(x, t),
∂t2
∂x2
u(0, t) = u(π, t) = 0,
∞
X
1
u(x, 0) =
sin nx,
2
n
n=1
0 < x < π,
t > 0,
t > 0,
0 < x < π,
∞
X (−1)n+1
∂u
(x, 0) =
sin nx,
∂t
n
n=1
0 < x < π.
Problem 10.2.28 (p.614) Show that if u(x, t) = X(x)T (t) is a solution of the partial
differential equation
2
∂ 2 u ∂u
2∂ u
+u=α
+
,
∂t2
∂t
∂x2
then X(x) and T (t) must satisfy the following ordinary differential equations:
X 00 (x) − λX(x) = 0,
T 00 (t) + T 0 (t) + (1 − λα2 )T (t) = 0,
where λ is a constant.
Problem 10.2.30 (p.614) Show that if u(r, θ, z) = R(r)T (θ)Z(z) is a solution of the partial
differential equation
1 ∂ 2u ∂ 2u
∂ 2 u 1 ∂u
+
+
+ 2 = 0,
∂r2
r ∂r r2 ∂θ2
∂z
then R(r), T (θ), and Z(z) must satisfy the following ordinary differential equations:
T 00 (θ) + µT (θ) = 0,
Z 00 (z) + λZ(z) = 0,
r2 R00 (r) + rR0 (r) − (r2 λ + µ)R(r) = 0,
where λ and µ are constants.