Abstract questions about PDEs with respect to Seperation of Variables

[trap101]

I have two more loosely based questions about PDEs and the separation of variables technique:


In the intro of this chapter the author imposed that we "assume" the the solution to a set of special PDEs is:

U(x,t) = X(x)T(t) where X and T are the eigenfunctions. My question is how did they derive that this form would work? I mean working from that equation you see it works, but how did they even conjure up the direction in which to go to arrive at that? This is my first PDE course so maybe that is above my pay grade and I should worry about that after.

The second question was how do I know that separation of variables is the right technique to use when solving a question, if I have other techniques to use?

Thanks

 

[phyzguy]

I have two more loosely based questions about PDEs and the separation of variables technique:

In the intro of this chapter the author imposed that we "assume" the the solution to a set of special PDEs is:

U(x,t) = X(x)T(t) where X and T are the eigenfunctions. My question is how did they derive that this form would work? I mean working from that equation you see it works, but how did they even conjure up the direction in which to go to arrive at that? This is my first PDE course so maybe that is above my pay grade and I should worry about that after.

The second question was how do I know that separation of variables is the right technique to use when solving a question, if I have other techniques to use?

Thanks

In fact, most of the time separation of variables doesn't work. But since it does work sometimes, and is relatively simple to do, it is usually worth a try. The thinking is as follows:

(1) Assume there is a solution that meets U(x,t) = X(x)T(t) .
(2) Convert the PDE to ODE's for X and T.
(3) Solve the ODE's for X and T.

If you can simultaneously solve the ODE's for X and T, then U is a solution to the original PDE. If you can't (usually you can't), then separation of variables didn't work and you have to try another method.

There is no general solution for solving PDE's, but a large number of techniques known to work in certain cases.