- #1
jaejoon89
- 195
- 0
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.
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.