Let $\mathcal{L}$ be a linear second-order differential operator on the interval $[0, \pi / 2]$. Consider the problem

$$\mathcal{L} y(x)=f(x) ; \quad y(0)=y(\pi / 2)=0$$

with $f(x)$ bounded in $[0, \pi / 2]$.

(i) How is a Green's function for this problem defined?

(ii) How is a solution $y(x)$ for this problem constructed from the Green's function?

(iii) Describe the continuity and jump conditions used in the construction of the Green's function.

(iv) Use the continuity and jump conditions to construct the Green's function for the differential equation

$$\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}+\frac{5}{4} y=f(x)$$

on the interval $[0, \pi / 2]$ with the boundary conditions $y(0)=0, y(\pi / 2)=0$ and an arbitrary bounded function $f(x)$. Use the Green's function to construct a solution $y(x)$ for the particular case $f(x)=e^{x / 2}$.