Differential Equation with Delta Dirac

Question

This is my first question, and it was my last solution, since no article could help me solve this differential equation.

The equation is in the following form:

$$\dfrac{d^2 f(x)}{dx^2}-Af(x)+B\delta(x-C)f(x) = 0 \quad x \in [0,L]$$ where $$\delta(x-C)= \infty\quad if \quad x=C$$or$$ \delta(x-C)=0 \quad if\quad x\neq C$$

What I'm Asking is the solution of $f(x).$

Ignoring the delta results in Exponential solutions, but delta function makes it difficult to calculate $f(x)$

P.S. : Had Kronecker instead of Dirac, which was TOTALLY wrong, that's why the 1st comments are kind of "strange" now.

Answer 1

Note that $\delta(x-C) \, f(x) = f(C) \, \delta(x-C)$ so the equation can be written $$f''(x) - A f(x) = -B f(C) \, \delta(x-C). \tag{1}\label{1}$$

Therefore solve $f''(x) - A f(x) = 0$ on the two intervals $[0,C)$ and $(C,L]$ and then "connect" the solutions so that $f$ is continuous but $f'$ has a step at $C$ such that $\eqref{1}$ is satisfied.

Answer 2

Perhaps this is meant to be a sort of "singular potential" problem? That is, it's intended to be equivalent to something like $(\Delta-\lambda)u=\delta_c$ on an interval containing $c$? Assuming so, this equation is readily solved by Fourier transforms.