solve the next differential equation:

$\displaystyle y´´- a*y= \delta (x-d)$

with the boundary conditions:

$\displaystyle \left.\frac{\partial y}{\partial x} \right|_ {x=0} = 0$

$\displaystyle y(x)\rightarrow 0 \hbox{ as } x\rightarrow \infty.$

I get the homogeneous solution: $\displaystyle y_H = C_1 exp (\sqrt{a}x) + C_2 exp (-\sqrt{a}x)$

Now, I have to obtain two solutions $\displaystyle u_1(x)$ and $\displaystyle u_2(x)$ which satisfy one of the two boundary conditions.

I have done the next:

1. $\displaystyle y=0 \rightarrow y_H = C_1 + C_2 \rightarrow C_1 = C_2 \rightarrow $

$\displaystyle C_1 [exp (\sqrt{a}x) + exp (-\sqrt{a}x)] = cosh (\sqrt{a}x) = u_1(x) $

2. $\displaystyle y= \infty \rightarrow y_H = C_2 exp (-\sqrt{a}x)= 0 \rightarrow u_2(x)=exp (-\sqrt{a}x)$

But I am not sure if this is correct.