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I am taking first course in PDEs and the only way i know of solving PDEs is separation of variables , and all the equations i saw had unique answers due to the ICs and BCs , but not this one : $$ \Delta u + u=0 , u(x,0) = u(x,b) = u(0,y) = u(a,y) = 0 $$ i was pretty sure that the only answer is $ u = 0 $ , but there are infinite number of answers , so my real question is : when does having $N$ conditions on $x$ , $M$ on $y$ , ... in a PDE consisting of $\frac{\partial^nu}{\partial x^n}$ , $\frac{\partial^mu}{\partial y^m}$ , ... is sufficient for a unique answer ?

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$\Delta u + u = 0$ is a Helmholtz equation, look it up. Or when we view it as $\Delta u = -1 \cdot u$, we see that we are searching for the eigenfunction of the Laplacian operator corresponding to eigenvalue -1.

For the uniqueness of PDE solutions in general, start reading here.

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