I need to solve following differential equation
\begin{eqnarray} y''(x) - k = \delta(x-x_0) \end{eqnarray}
subject to conditions: \begin{eqnarray} y(x=-a) = 0 \\ y(x=b) = p \end{eqnarray}
Is it possible to solve the equation using Green's function? I tried as follows: solution of the homogeneous equation \begin{eqnarray} y''(x) - k = 0 \end{eqnarray}
is $y(x) = k/2x^2 + bx + c$
but in this case I am not able to split $y(x)$ into two liner independent functions to use boundary conditions. How can I proceed?
EDIT
Following Silynn's comment This seems to be the solution: \begin{eqnarray} y(x) &=& (x-x_0)\theta(x-x_0) + \frac{k}{2}x^2 + a_1x + a_2 \\ y'(x) &=& (x-x_0)\delta(x-x_0) + \theta(x-x_0)+ kx + a_1 \\ y''(x) &=& \delta(x-x_0)+ k \\ \end{eqnarray}