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Suppose we have a PDE and we can solve it with the separation of variables and let's say the variables are $x,y,z$. Is the factorized function $f(x,y,z)=g(x)h(y)l(z)$ the only possible solution? Are there any solutions of a different form other than $f(x,y,z)$?

Boar
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Luigid
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  • The superposition principle might get you more solutions. Are you familiar with linear algebra? – Boar May 20 '22 at 10:26
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    @Steve Yes, I'm a bit familiar. Since the PDE is also a linear ED, even a linear combination of particular solutions is a solution. Do you mean this? – Luigid May 20 '22 at 13:04
  • For a homogeneous equation, yes, that's what I mean. Actually, you may need the notion of bases for a solution space to crack the problem. I'm sorry to tell you that I've been away from differential equations for a long time, and I recommend you the book written by Boyce and DiPrima. This book might include something useful. Hope that helps. – Boar May 21 '22 at 01:01
  • The sum of factored functions is not generally factorable. – Disintegrating By Parts May 21 '22 at 22:01

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