Try these references:

In general, analytical solution for PDE with variable coefficients is more an exclusion than a typical case.

I think that this lecture can be a good introduction to the problem: http://www.math.ucsb.edu/~grigoryan/124A.pdf

First you can see classification: (1.2)-(1.8). Each PDE has its own properties and require different approach. Elliptic equations differ from hyperbolic that often have solutions in a form of waves.

For hyperbolic equations one can use the method of characteristics: http://www.mathphysics.com/pde/herod/jvh13.html

If the coefficients vary slowly, there are also asymptotic methods that give a solution in a form of series in powers of a small parameter. This is not an easy calculation. You can find an example of its application to one PDE of his type in one of my papers (where derivation is also presented): https://www.researchgate.net/publication/275581998_Evolution_of_long_nonlinear_waves_on_shelves

However, this equation is more complicated; it is a nonlinear PDE with variable coefficients.

Conference Paper Evolution of long nonlinear waves on shelves

Dear Pranav, 

I suppose that  

Hans Triebel's book, Higher Analysis. Barth, Leipzig/Berlin/Heidelberg 1992, ISBN 3-335-00321-7, could be a good reference. 

Best regards, 

Jürgen 

Some good references that can be used are books of Lars Hormander.

The following references may be relevant

http://dx.doi.org/10.1016/j.cnsns.2009.01.023

http://dx.doi.org/10.1016/j.cnsns.2010.01.037  (http://arxiv.org/abs/0911.1848)

http://dx.doi.org/10.1088/0031-8949/89/03/038003   (http://arxiv.org/abs/1308.5126)

For the beam equation, see, e.g.,

http://dx.doi.org/10.1088/1751-8113/41/13/135206