I am trying to find the vibration modes of a string that has a uniform mass density, plus some point mass somewhere attached to it, modelled by an additional Dirac delta function in the mass density. The wave equation is of the form
$u_{xx}=(1+\delta{(x)})u_{tt}$,
where $u$ is the deformation, and $(1+\delta{(x)})$ the mass density. After separation of variables we find
$X_{xx}=(1+\delta{(x)})X$,
where $X$ is the spatial part of the solution. Is there any analytical solution for $X$?
We want to solve the differential equation $y''(x) = (1+\delta(x)) y(x),$ where $\delta$ is the Dirac delta "function".
Since $f(x)\delta(x) = f(0)\delta(x)$ for every $f$ continuous at $x=0,$ the equation reduces to $y''(x) = y(x) + y(0)\delta(x),$ i.e. $y''(x)-y(x)=y(0)\delta(x).$ A solution to this is continuous, satisfies $y''-y=0$ on $(-\infty,0)$ and on $(0,\infty),$ and its derivative makes a jump at $x=0$ which is equal to $y(0).$
Let $y(x)=Ae^{x} + Be^{-x}$ be the solution on $(-\infty, 0)$ and $y(x)=Ce^{x} + De^{-x}$ be the solution on $(0, \infty).$ Then, for continuity we shall have $A+B=C+D=:E,$ and to get $y''-y=y(0)\delta$ we shall have $(C-D)-(A-B)=y(0)=E.$
Now you only have to find all $A,B,C,D \in \mathbb{R}$ such that $A+B=C+D=(C-D)-(A-B).$