Question: Following the proof of (12.4), show that (12.9) gives a solution of (12.7).

The given expressions are $y\left(x\right)={\int }_0^{\frac{\pi }{2}}G\left(x,x^{\text{'}}\right)f\left(x^{\text{'}}\right)d{x}^{\text{'}}$.

In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.

$y^{\text{'}\text{'}}+y=f\left(x\right)$

We first wish to find the solution to the unit excitation which is given by a Green function:

$\frac{d^2}{d{x}^2}G\left(x,x^{\text{'}}\right)+G\left(x,x^{\text{'}}\right)=\delta \left(x^{\text{'}}-x\right)$

We now wish to impose the condition that $y(0)=y(\pi / 2)=0$. If we assume the solution of the form:

$y\left(x\right)={\int }_0^{\pi /2}G\left(x,x^{\text{'}}\right)f\left(x^{\text{'}}\right)d{x}^{\text{'}}$

We know that Green's function satisfies:

$\frac{d^2}{d{x}^2}G\left(x,x^{\text{'}}\right)+G\left(x,x^{\text{'}}\right)=\delta \left(x-x^{\text{'}}\right)$

Inserting assumed solution yields:

$y^{\text{'}\text{'}}+y=\left(\frac{d^2}{d{x}^2}+1\right)y=\left(\frac{d^2}{d{x}^2}+1\right){\int }_0^{\pi /2}G\left(x,x^{\text{'}}\right)f\left(x^{\text{'}}\right)d{x}^{\text{'}}$

We know that:

$\begin{array}{l}\frac{d^2}{d{x}^2}G\left(x,x^{\text{'}}\right)+G\left(x,x^{\text{'}}\right)=\delta \left(x-x^{\text{'}}\right)\\ \left(\frac{d^2}{d{x}^2}+1\right)G\left(x,x^{\text{'}}\right)=\delta \left(x-x^{\text{'}}\right)\end{array}$

Inserting this gives:

We know that:

$\begin{array}{l}\frac{d^2}{d{x}^2}G\left(x,x^{\text{'}}\right)+G\left(x,x^{\text{'}}\right)=\delta \left(x-x^{\text{'}}\right)\\ \left(\frac{d^2}{d{x}^2}+1\right)G\left(x,x^{\text{'}}\right)=\delta \left(x-x^{\text{'}}\right)\end{array}$

Inserting this gives:

$y^{\text{'}\text{'}}+y=-{\int }_0^{\pi /2}\delta \left(x-x^{\text{'}}\right)f\left(x^{\text{'}}\right)d{x}^{\text{'}}=f\left(x\right)$

To satisfy the boundary conditions, one would need to find a general form of the Green function and set parameters in a way that satisfies all the required conditions.