Question

See next comment, question on PDE separation of variables.

Answer 1

This a portion of a larger problem i'm working on. This comes from solving the 1-D heat equation with certain boundary/initial conditions. Here I'm stuck on finding eigenvalues/functions that are non-trivial solutions to X(x).

The heat equation is:\[u_t=u_{xx}\]Boundary Conditions:\[u(0,t)=0\]\[u_x(\pi,t)=-u(\pi,t)\]

After separation\[X''(x)+\lambda X(x)=0\]

\[\lambda=0\]\[X(x)=Ax+b\]
Note: mu = -lambda
\[\lambda <0\]\[X(x)=Acosh(\sqrt{\mu}x)+Bsinh(\sqrt{\mu}x)\]
\[\lambda>0\]\[X(x)=Acos(\sqrt{\lambda}x)+Bsin(\sqrt{\lambda}x)\]

For all three cases I found that all coefficients were 0, thus there are no non-trivial solutions. I don't think this is right?

Answer 2

let me exactly solve it.