Chapter 13.9, Problem 3P

Consider the heat flow problem of Section 3. Solve this by Laplace transforms (with respect to $t$ ) by starting as in Example 1. You should get $\frac{{\partial }^{2}U}{\partial x^2}-\frac{p}{{\alpha }^2}U=-\frac{100}{{\alpha }^{2}l}x\text{ and }U(0,p)=U(l,p)=0$

Solve this differential equation to get $U(x,p)=-\frac{100\sinh \left(p^{1/2}/\alpha \right)x}{p\sinh \left(p^{1/2}/\alpha \right)l}+\frac{100}{pl}x.$

Assume the following expansion, and find $u$ by looking up the inverse Laplace transforms of the individual terms of $\text{U}$ :

Your answer should be (3.15).