Chapter 10.7, Problem 17P

a.

To determine

To show: The equation $\varphi \left(x\right)+\psi \left(x\right)=0$ and $-a{\varphi }^{\prime }\left(x\right)+a{\psi }^{\prime }\left(x\right)=g\left(x\right)$ by using the form of solution obtained in Problem 13.

b.

To determine

To show: The equation ${\psi }^{\prime }\left(x\right)=-{\varphi }^{\prime }\left(x\right)$ by using the first equation of part a.; show $-2a{\varphi }^{\prime }\left(x\right)=g\left(x\right)$ which implies that $\varphi \left(x\right)=-\frac{1}{2a}{\displaystyle \underset{x_0}{\overset{x}{\int }}g\left(\xi \right)}d\xi +\varphi \left(x_0\right)$ where $x_0$ is an arbitrary point then determine $\psi \left(x\right)$, by using the second equation.

c.

To determine

To show: The equation $u\left(x,t\right)=\frac{1}{2a}{\displaystyle \underset{x-at}{\overset{x+at}{\int }}g\left(\xi \right)}d\xi$.