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.
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.


We choose to use spherical coordinates (in your example) precisely because we want to examine situations which have spherical geometries.

We could equally choose e.g. cylindrical coordinates to study systems with cylindrical geometries.  If you search for this you'll see cylindrical coordinates crop up quite often  in physical models.

Also, there are very few equations (or forms of equation) which admit simple closed form solutions or even the ability to decouple using separation of variables.  When we get a chance to use the technique, we grab it. :-)

Often a good choice of coordinate system will allow useful exploration of a problem even if an explicit solution remains out of reach.
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.


We choose to use spherical coordinates (in your example) precisely because we want to examine situations which have spherical geometries.

We could equally choose e.g. cylindrical coordinates to study systems with cylindrical geometries.  If you search for this you'll see cylindrical coordinates crop up quite often  in physical models.

Also, there are very few equations (or forms of equation) which admit simple closed form solutions or even the ability to decouple using separation of variables.  When we get a chance to use the technique, we grab it. :-)

Often a good choice of coordinate system will allow useful exploration of a problem even if an explicit solution remains out of reach.