Two Partial Differential Equations

[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.

 

[paranormal]

As a scientist, my response to this content would be to carefully review the steps taken to reach this solution and make sure that the separation of variables method was applied correctly. I would also suggest checking the boundary conditions and verifying that the chosen lambda value is appropriate for the problem. Additionally, I would recommend considering alternative methods for solving the partial differential equations, such as using Green's functions or numerical methods. It is important to thoroughly understand and validate any solutions in order to confidently apply them to real-world problems.