When can seperation of variables be applied?

PhDorBust
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I have some confusion about when separation variables can be applied to a PDE. Can it be applied on any PDE that can separated for any domain? If so, is the use of more "powerful" techniques simply used to save effort? (As you might have to use superposition several times!)
 
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To my knowledge, it can be use in a square domain only(i.e. x1 in [a,b] and x2 in [c,d]).
Non-square domain is usually transformed into square domain by the change of variable, in order to separate the variables.
 
Thread 'Direction Fields and Isoclines'

Thread 'Direction Fields and Isoclines'

I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
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