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For conventional PDEs like diffusion, waves, it seems the standard way to solving them is in two steps.

1. Use separation of variables method to make them into ODEs

2. Use eigenvalues and eigenfunctions theory on ODEs to construct the final solution consisting of an infinite number of eigenfunctions which statisfies the BC and intintial conditions. Some people say that to solve them you use either separation of variables technique or eigen theory. But to me they are intimately related and

1. Use separation of variables method to make them into ODEs

2. Use eigenvalues and eigenfunctions theory on ODEs to construct the final solution consisting of an infinite number of eigenfunctions which statisfies the BC and intintial conditions. Some people say that to solve them you use either separation of variables technique or eigen theory. But to me they are intimately related and

**both**are needed in solving PDEs. Am I correct?
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