Are You on the Right Track with Separation of Variables for PDEs?

[Kidphysics]

Homework Statement

This is the first problem of the two.

Homework Equations

The Attempt at a Solution

Using separation of variables, I end up with

T'(t)= -λKT(t) and X''(x)+(β/K)X'(x)/X(x)= -λ. At first I chose the negative lambda because I saw that U(0,t) and U(L,t) needed to oscillate and I was hoping to get a sin function. Now the characteristic equation for X is something like r^2 + (β/K)r +λ=0 and I am not sure if I am on the right track in solving for the function X(x).

 

[frogjg2003]

You're on the right path, but it helps to use ² instead of λ.

 

[Kidphysics]

thanks. I guess what I should have said is that I am stuck at this point.

 

[frogjg2003]

Solve the characteristic equation for r. The solution has the form C₁Exp(rx)+C₂Exp(-rx).
This might become D₁sin(r'x)+D₂cos(r'x) depending on the sizes of β, k, and λ.
Edit:If r was imaginary, r' is a new constant that is real.