I am reading "Topics of Functional Analysis and Applications" by Kesavan by myself. This is an exercise from the first chapter.
Consider the differential operator $\Big( \dfrac{d^2}{dx^2} + a \dfrac{d}{dx} + b \Big)=L$ , $a,b$ are constants. Let $f,g$ satisfy: $Lf=0, Lg=0 , f(0)=g(0), f'(0)-g'(0)=1.$
Consider the function, $\begin{equation} F(x) = \begin{cases} f(x) \ \ \ \text{when} \ x\leq 0\\ g(x) \ \ \ \text{when} \ x >0 \end{cases} \end{equation}$
Show that $-F$ is a Fundamental solution for $L$.
I have no idea how to approach such problems to find the fundamental solution or how to prove something is the fundamental solution, I know I have to show $L(-F)=\delta$. Thanks in advance for helping.