Chapter 13.2, Problem 15P

Consider a finite plate, $10\text{cm}$ by $30\text{cm},$ with two insulated sides, one end at $0^{°}$ and the other at a given temperature $T=f(x).$ Try $f(x)={100}^{°};f(x)=x.$ You should convince yourself that this problem cannot be done using just the solutions (2.7)To see what is wrong, go back to the differential equations ( 2.5 ) and solve themif $k=0.$ You should find solutions x, y, xy and constant [the constant is already contained in (2.7) for k=0, but the other three solutions are not]. Now go backover each of the problems we have done so far and see why we could ignore these $k=0$ solutions; then including the $k=0$ solutions, finish the problem of the finite plate with insulated sides.

For the case $f(x)=x,$ the answer is: $T=\frac{1}{6}(30-y)-\frac{40}{{\pi }^2}{\displaystyle \underset{\text{odd }n}{\sum }\frac{1}{n^2\sinh 3n\pi }}\sinh \frac{n\pi }{10}(30-y)\cos \frac{n\pi x}{10}$.