Consider the Laplace equation $\nabla^2 u=0$. We can find a set of solutions for that by assuming $u=f(x)g(y)$. Also we can find another set of solutions by assuming $u=f(x)+g(y)$ that is not the same as the first set. Which of these solution are correct and if both are correct why a partial differential equation has two or more sets of the solutions?
Thank you
By their nature differential equations have infinitely many solutions. Even a simple ODE as $y'=f(x)$ has a whole one-parameter family of solutions. A PDE has even more freedom. Instead of just a finite number of parameters a PDE has solutions depending on arbitrary functions, that can be determined via initial/boundary conditions. For ex something like $\partial y/\partial x=0$ for a function $y=y(x,t)$ has a solution depending on an arbitrary function of $t$. You can try many different ansatz of solutions, with multiplicative or additive structure and you find solutions of different types with different symmetry.