Is this equation exactly solvable?

Question

I would like to know, if the following equation (which resembles the Sturm-Liouville equation) is exactly solvable (and how, if yes): $$\frac{d}{dx}\left[p(x)\frac{d}{dx}y(x)\right] - k^2p(x)y(x) = 0.$$

Thank you for your help!

Answer 1

You could do this with a series of quadratures. $$ \frac{d}{dx}\left[p(x)\frac{d}{dx}y(x)\right] = k^2 p(x)y(x) \\ p(x)\frac{d}{dx}y(x) = \int_0^x k^2p(x')y(x')dx'+A \\ \frac{d}{dx}y(x) = \frac{1}{p(x)}\int_0^xk^2 p(x')y(x')dx'+A\frac{1}{p(x)} \\ y(x)=\int_0^x\frac{1}{p(x'')}\int_{0}^{x''}k^2p(x')y(x')dx'dx''+A\int_0^{x}\frac{1}{p(x')}dx'+B $$ You can start with $y=0$ and begin the process of iterating and end up with a series solution with arbitrary constants $A,B$.