Chapter 13.7, Problem 9E

To determine

To show: The steady state temperature $u\left(x,t\right)$ of unit length rod is $u\left(x,t\right)={\displaystyle \sum _{n=1}^{\infty }{A}_n{e}^{-k{\alpha }_n^{2}t}\left({\alpha }_n\cos \left({\alpha }_{n}x\right)+h\sin \left({\alpha }_{n}x\right)\right)}$ where, $A_n=\frac{2}{\left({\alpha }_n^2+2h+h^2\right)}{\displaystyle {\int }_0^{1}f\left(x\right)\left(\alpha \cos \left({\alpha }_{n}x\right)+h\sin \left({\alpha }_{n}x\right)\right)dx}$.