DiffEq question (solving damped wave equation)

[jaejoon89]

How do you use separation of variables to solve the damped wave equation
y_tt + 2y_t = y_xx

where y(0,t) = y(pi,t) = 0
y(x,0) = f(x)
y_t (x,0) = 0

---
These are partial derivatives where y = X(x)T(t)

So rewriting the equation I get

X(x)T''(t) + 2X(x)T'(t) = X''(x)T(t)
which results in the following differential equations (lambda a constant)

X''(x) + lambda*X(x) = 0...(1)
T''(t) + 2T'(t) + lambda*T(t) = 0...(2)

I am supposed to use the boundary conditions somehow and find y(x,t) which will be a Fourier series but I am completely stuck and would appreciate some help.

 

[HallsofIvy]

How do you use separation of variables to solve the damped wave equation
y_tt + 2y_t = y_xx

where y(0,t) = y(pi,t) = 0
y(x,0) = f(x)
y_t (x,0) = 0

---
These are partial derivatives where y = X(x)T(t)

So rewriting the equation I get

X(x)T''(t) + 2X(x)T'(t) = X''(x)T(t)
which results in the following differential equations (lambda a constant)

X''(x) + lambda*X(x) = 0...(1)
T''(t) + 2T'(t) + lambda*T(t) = 0...(2)

I am supposed to use the boundary conditions somehow and find y(x,t) which will be a Fourier series but I am completely stuck and would appreciate some help.

You are told that y(0, t)= X(0)T(t)= 0 and that y(pi, t)= X(pi)T(t)= 0. In order that those be true for all t, you must have X(0)= 0 and X(pi)= 0.

Start by solving the equation X"+ lambda X= 0, X(0)= 0, X(pi)= 0.

The "type" of soltutions will (1) linear if lambda= 0, (2) exponential if lambda< 0, (3) sine and cosine if lambda> 0. But you know that X(0)= X(pi)= 0 so lambda must be what?