How can one proceed for solving differential equations in physics without separation of variables? For ex- Take Laplace equation in spherical coordinates, we always assume the solutions of form R(r)Θ(θ)Φ(φ) and then we resolve the differential equation in three differential equations of single variable. Doesn't it restrict the type of solutions. What if other solutions cannot be written in R(r)Θ(θ)Φ(φ) form, how do we find such solutions then.
I have same doubt for other differential equations of central importance in physics. "Schrodinger equation for hydrogen atom", "Wave-equation" "Diffusion-equation" and many more.