Partial Differential Equation - solve with Matlab

[Dmitrij00000]

Help please, I need to solve this differential equation \(\displaystyle x\frac{\partial^2 U}{\partial x^2}+y\frac{\partial^2 U}{\partial y^2}=aU\) in Matlab (where "a" is a constant parameter, it can be taken by any), I wanted to use the Partial Differential Equation Toolbox, but I ran into a problem, the elliptic equation in this Toolbox is represented in a vector form, namely -div(c*grad(u))+a*u=f.
Please help me convert my equation to this form and tell me how it can be done or at least name the sources of information from which I can learn this knowledge.
I really need to solve this equation in Matlab,so please tell me how it can be done.

 

[gtoguy97]

The general solution of the differential equation $x\frac{\partial^2 U}{\partial x^2}+y\frac{\partial^2 U}{\partial y^2}=aU$ can be expressed as a linear combination of the two-dimensional Fourier sine and cosine series. This can be written as $U(x,y)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}A_{mn}\cos(mx)\cos(ny)+B_{mn}\sin(mx)\sin(ny)$.The coefficients $A_{mn}, B_{mn}$ can be determined using the boundary conditions. In order to solve this equation using the Partial Differential Equation Toolbox in Matlab, we need to convert it into the form -div(c*grad(u))+a*u=f. This can be done by multiplying both sides of the equation by the function $U$ and then integrating with respect to $x$ and $y$. This gives us $\int_0^{2\pi}\int_0^{\infty}U(x,y)\left[x\frac{\partial^2 U}{\partial x^2}+y\frac{\partial^2 U}{\partial y^2}\right]dxdy=\int_0^{2\pi}\int_0^{\infty}aU^2dxdy$.Integrating by parts on the left hand side, we get$-\int_0^{2\pi}\int_0^{\infty}\nabla U(x,y)\cdot\nabla U(x,y)dxdy+\int_0^{2\pi}\int_0^{\infty}U(x,y)\frac{\partial U(x,y)}{\partial x}dxdy=\int_0^{2\pi}\int_0^{\infty}aU^2dxdy$.Now we can rewrite this equation in the form -div(c*grad(u))+a*u=f where c is the identity matrix and f is zero. This will give us $-div(\nabla U(x,y))+aU(x,y