Chapter 10.2, Problem 26E

Verify that $u_n\left(x,t\right)$ given in equation (10) satisfies equation (1) and the boundary conditions in (2) by substituting $u_n\left(x,t\right)$ directly into the equations involved.

$\begin{array}{l}{u}_n\left(x,t\right)=X_n\left(x\right)T_n\left(t\right)=a_n\sin \left(n\pi x/L\right)b_n{e}^{-\beta {\left(n\pi /L\right)}^{2}t}\left(10\right)\\ \frac{\partial u}{\partial t}\left(x,t\right)=\beta \frac{{\partial }^{2}u}{\partial x^2}\left(x,t\right),0<x<L,t>0\left(1\right)\\ u\left(0,t\right)=u\left(L,t\right)=0,t>0,\left(2\right)\end{array}$