I am trying to tackle a second order(also of two variables), linear pde with variable coefficients. I don't want to post the problem because I would prefer to solve it myself, but the coefficients are all periodic. Additionally, the discriminant of the PDE can be positive, negative, or zero.
I was hoping to use the method of characteristics to solve this, but I'm not sure how to properly do this when the discriminant changes sign. I can't convert this to any canonical form because of this.
Is it possible to break up the domain into separate regions based on the sign of the discriminant and change it to canonical form in each one?