Method of separation of variables (another)

Question

Given an ODE or a PDE one can sometimes use the method of separation of variables. If so, are the solutions obtained via this method general and always valid?

In other words, can one use always this method or are there some conditions on using it?

Answer 1

The method of separation of variables gives particular solutions : Only the solutions on a particular form (the form chosen to make the variables separated).

Of course, the particular functions obtained are solutions of the PDE, but they are far to be all the solutions. In case of linear homogeneous PDE, other solutions are obtained on the form of any linear combination of the previous particular solutions, with arbitrary coefficients.

Answer 2

Separation of variables means different things in ODE and in PDE. Let me show an example of each and discuss generality.

In ODE: Imagine the simple equation $$\frac{dy}{dx}=\frac{x}{y} $$ To separate variable we put all the xs on one side all the ys on the other (this is not always possible) and we get $$ydy=x dx$$ We integrate and get $$\frac{1}{2}y^2 =\frac{1}{2}x^2+C$$ Given proper BC we will have a unique solution (by existence and uniqueness theorem of ODE). This solution is general (no assumptions were made to obtain it). It is however worth noting that this only works for

For a PDE, separation of variables means assume your solution $u(x,t)=f(x)g(t)$ or a linear combination of such solutions. This is not always true (for example, even if $u=\sin(xt)$ satisfied your equation, you would not find it with separation of variables). So the method of separation of variables only gives solutions of a certain form. However, I am unaware of a general method to finding the solutions to a PDE, all the methods I know involve guessing a form of the solution.