6 7.20m, §6 Total

Collecting all the boxed formulae together, the solution is found by first computing the coefficients $\bar f_n$ from:

$\begin{displaymath} \bar f_n = \frac2\ell {\int_0^\ell \bar f(x)\cos((2n-1) \pi x/2\ell)\d x} \qquad (n=1,2,3,\ldots) \end{displaymath}$

where

$\begin{displaymath} \bar f(x) = f(x) - g_1(0) - g_0(0)(x-\ell) \end{displaymath}$

Also compute the functions $\bar q_n(t)$ from:

$\begin{displaymath} \bar q_n(t) = \frac2\ell {\int_0^\ell \bar q(x,t)\cos((2n-1)\pi x/2\ell)\d x} \qquad (n=1,2,3,\ldots) \end{displaymath}$

where

$\begin{displaymath} \bar q(x,t) = q(x,t) -g_1'(t) - g_0'(t)(x-\ell) \end{displaymath}$

Then the temperature is:

$\begin{displaymath} \begin{array}{l} \displaystyle u(x,t)_{\strut} = g_1(t) +... ...\pi^2 t/4\ell^2} \right] \cos((2n-1)\pi x/2\ell) \end{array}\end{displaymath}$