Solving Linear ODEs over the space of distributions

Question

I am encountering in my work linear differential equations with coefficients that involve things like Heaviside functions and delta Dirac functions. I know how to find things like Green's functions or solve the Schrödinger equation involving a Dirac delta well. You solve on the left and right and get the jump condition. I am looking for something more systematic as my equations do not alway fit the nice mold of those examples. I found myself being able to somehow solve my equations but there must be a reference out there that gives systematic methods of solving equations in the space of distributions. A simple example of an equation involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the converging solution as $x\rightarrow \infty$, then solve on the left similarly. Assume continuity of the solution $y$. The jump condition on $y'$ is found by integrating from $-\epsilon$ to $\epsilon$ and take $\epsilon\rightarrow 0$. The only value of $\lambda$ giving a bound state is then found to be $1/2$. The example I am facing is a third order equation, which does not fits the mold of examples I know such as the one above.