[SOLVED] Amath 567, homework 6

1. Problem 1:
(a) Let ˆf(s) and ˆg(s) be the Laplace transforms of one-sided functions f(t) and g(t),
respectively. Show that the inverse Laplace transform of ˆf(s)ˆg(s) is
Z t
0
f(t − τ )g(τ )dτ(b) Use the Laplace transform and the result in (a) to solve the following ordinary
differential equation d
2y
dt2 + 4y = f(t), subject to the initial conditions y(0) = 0,
dy
dt (0) = 02. Problem 2: Solve the following Laplace equation
∂
2ϕ
∂x2
+
∂
2ϕ
∂y2
= 0
in the upper half plane x ∈ (−∞, ∞) y ∈ [0, ∞), subject to the boundary conditions
ϕ → 0 as y → ∞; ϕ → 0 as x → ±∞;
ϕ(x, 0) = x
x
2 + a
23. Problem 3: Use the Fourier transform to solve the following wave equation:
∂
2u
∂t2
= c
2 ∂
2u
∂x2
where x ∈ (−∞, ∞) and t ∈ [0, ∞), subject to the initial condition u(x, 0) = 0,
∂u
∂t (x, 0) = δ(x) and the boundary conditions u(x, t) → 0 as x → ±∞.