Chapter 13.9, Problem 4P

A semi-infinite bar is initially at temperature ${100}^{°}$ for $0<x<1,$ and $0^{°}$ for $x>1$ Starting at $t=0,$ the end $x=0$ is maintained at $0^{°}$ and the sides are insulated. Find the temperature in the bar at time t,as follows. Separate variables in the heat flow equation and get elementary solutions $e^{-{\alpha }^2{k}^{2}t}\sin kx$ and $e^{-{\alpha }^2{k}^{2}t}\cos kx$ Discard the cosines since $u=0$ at $x=0.$ Look for a solution. $u(x,t)={\displaystyle {\int }_0^{\infty }B}(k)e^{-{\alpha }^2{k}^{2}t}\sin kxdk$. and proceed as in Example 2 .Leave your answer as an integral.