Solving Differential Equations without separation of variables

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.

• Nobody said that you could not solve the Schrodinger equation for the hydrogen atom in (x,y,z) space. It just would be, well, harder. – Jon Custer Sep 27 '17 at 18:24
• Mathematics rather than Physics? – Farcher Sep 27 '17 at 18:25
• Farcher---> Yeah, it's related to mathematics but these differential equations are of central importance in physics. – Jitendra Sep 27 '17 at 18:38
• there's always the method of characteristics: en.m.wikipedia.org/wiki/Method_of_characteristics – user160660 Sep 27 '17 at 18:39
• Thank You very much- ZeroTheHero. – Jitendra Sep 27 '17 at 18:47

Take Laplace equation in spherical coordinates, we always assume the solutions of form $R(r)\Theta(\theta)\Phi(\phi)$ and then we resolve the differential equation in three differential equations of single variable. Doesn't it restrict the type of solutions.