When can I use separable solutions for PDE's?

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SUMMARY

Separable solutions for partial differential equations (PDEs) can be utilized when the function can be expressed as a product of functions of individual variables, specifically in the form y(x,t) = X(t)T(t). The ability to separate variables is contingent upon both the governing equation and the geometry of the domain, such as whether it is a sphere, circle, or rectangle. Furthermore, separation of variables is applicable exclusively under homogeneous boundary conditions, as noted in the professor's lecture materials.

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Winzer
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So when can and can't I use separable solutions for PDE's?
Domain Constraints?
 
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Rather than look for a general rule on when you can or cannot "separate" functions, it is better to try. Write the function y(x,t) as X(t)T(t). IF you are able to separate the the two functions and varibles, good.

The problem is that whether or not a problem is separable depends not only on equation but also the "geometry" (whether you are working inside a sphere or circle or rectangle) which determines the coordinate system.
 
Thanks Halls.

I just poured through my profs lecture notes and found: "Separation of variables only works for homogeneous boundary conditions."

Just in case anyone else is interested.
 

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