Is Separation of Variables Valid for Solving Partial Differential Equations?

[captain]

when is the separation of variables technique for partial differential equations valid? it seems to give a particular general solution (such as a general Fourier series, or series of legendre polynomials) to a problem depending which coordinate system that you are in?

 

[HallsofIvy]

Any time your equation is linear. Separation of variables works as long as it is possible to "disassemble" your equation, solve each part, then put them back together into a solution of the entire equation. That is basically what "linear" allows us to do.

 

[dhris]

Any time your equation is linear.

I think that's a little strong since not every linear partial differential equation is separable.

 

[John Creighto]

I think that's a little strong since not every linear partial differential equation is separable.

I wonder what the conditions are. Would all first order linear PDEs be separable? If there was a forcing function could we just deal with the homogeneous part like we can for ODEs?