Chapter 11.3, Problem 30P

a.

To determine

To verify: The Green’s function obtained in problem 28 is given by the formula, $G\left(x,s\right)=\{\begin{array}{l}-\frac{y_1\left(s\right)y_2\left(x\right)}{p\left(x\right)W\left[y_1,y_2\right]\left(x\right)},0\le s\le x\\ -\frac{y_1\left(x\right)y_2\left(s\right)}{p\left(x\right)W\left[y_1,y_2\right]\left(x\right)},x\le s\le 1\end{array}$.

b.

To determine

To verify: The Green’s function obtained in problem 29 is given by the formula, $G\left(x,s\right)=\{\begin{array}{l}-\frac{y_1\left(s\right)y_2\left(x\right)}{p\left(x\right)W\left[y_1,y_2\right]\left(x\right)},0\le s\le x\\ -\frac{y_1\left(x\right)y_2\left(s\right)}{p\left(x\right)W\left[y_1,y_2\right]\left(x\right)},x\le s\le 1\end{array}$.

c.

To determine

To show: The expression $p\left(x\right)W\left[y_1,y_2\right]\left(x\right)$ is a constant by showing $\frac{d}{dx}\left(p\left(x\right)W\left[y_1,y_2\right]\left(x\right)\right)=0$.

d.

To determine

To show: The expression $G\left(x,s\right)=G\left(s,x\right)$ using the equation $G\left(x,s\right)=\{\begin{array}{l}-\frac{y_1\left(s\right)y_2\left(x\right)}{p\left(x\right)W\left[y_1,y_2\right]\left(x\right)},0\le s\le x\\ -\frac{y_1\left(x\right)y_2\left(s\right)}{p\left(x\right)W\left[y_1,y_2\right]\left(x\right)},x\le s\le 1\end{array}$.

e.

To determine

To show: The function $y=\varphi \left(x\right)$ from eq. (60) and $G\left(x,s\right)=\{\begin{array}{l}-\frac{y_1\left(s\right)y_2\left(x\right)}{p\left(x\right)W\left[y_1,y_2\right]\left(x\right)},0\le s\le x\\ -\frac{y_1\left(x\right)y_2\left(s\right)}{p\left(x\right)W\left[y_1,y_2\right]\left(x\right)},x\le s\le 1\end{array}$ satisfies the Sturm-Liouville problem and it’s boundary conditions.