This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f(x)$ throughout and if the ends $x=0$ and $x=L$ are insulated. $F(x)= x, 0$
Solution:
For an insulated rod the solution $ X(x,t)= \frac{a_0}{2}+\sum B_0 \frac{\cos(n\pi)x}{Le}−\frac{n^2\pi^2\alpha^2}{L}t $ I found $a_0= 1$ and $ B_n= −\frac{2}{n\pi}\sin\left(\frac{n\pi}{2}\right)+\frac{2}{n\pi}2\cos\left(\frac{n\pi}{2}\right)−\left(\frac{2}{n\pi}\right)^2 $ then just plug in the coefficients into the sum. I am just not sure if these are the correct values for the $a_0$ and $B_0$ coefficients.