# Linear second order partial differential equation with variable coefficients

So after using the method of separation of variables of a PDE in 3 variables, I have simplified it to a ODE in time and a PDE of two variables. However, my resulting PDE is a linear second order partial differential equation with variable coefficients of the form

$$Au_{xx} + Bu_{yy} + C(x)u_x + D(y)u_y + (p(x,y) + E)u = 0$$

where A,B, and E are constants, p(x,y) is a polynomial of x and y of degree 2, and C(x) and D(y) are functions of x and y respectively. Any suggestions on how to solve the resulting PDE?

• Do you have any reason to think that the values of $A$ and $B$ are such that your PDE is elliptic ? This is probably an easier framework to work with. – Gâteau-Gallois Nov 16 '18 at 8:29
• Yes, definitely. I forgot to mention it, but the PDE is elliptic on a bounded rectangle (of some sort). I was wondering if I am able to solve is using the separation of variables again with u = v(x)w(y)? If doing so I end up with a ODE for x on one side and an ODE for y on the other, but they are both depending on the common polynomial p(x,y). @Gâteau-Gallois – Ken Klark Nov 16 '18 at 14:53