How to numerically solve the eigenvalues of a partial differential operator？

Question

I have a partial differential operator of the following form: $$L_X=A(X) \frac{\partial}{\partial X} + \frac{1}{2}D(X)\frac{\partial^2}{\partial X^2}$$ where A and B allow to be nonlinear.For example: $$A(X)=\begin{pmatrix} ax+by+x^2+y^2 \\ cx+dy+x^2+y^2 \end{pmatrix}$$ Is there any way to numerically calculate the eigenvalues of this operator $\lambda$： $$L_Xu(X)=\lambda u(X)$$ I've seen many questions with partial differential operators, but they are generally linear, and I don't know how such an equation should be done.