Solve the heat equation

$\displaystyle k\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}, 0< x < L, t > 0$

$\displaystyle u(0,t) = 0, u(L,t) = 0, t > 0$

$\displaystyle u(x,0) = \begin{cases} x, &0 < x < \frac{L}{2}\\ L-x, &\frac{L}{2}

using

$\displaystyle u(x,t) = \frac{2}{L}\displaystyle\sum\limits_{n=1}^\infty \left( \int_0^L u(x,0)\sin\left( \frac{n\pi}{L}x\right) dx\right) e^{\left( -k\frac{n^2\pi^2}{L^2}t\right)} \sin\left( \frac{n\pi}{L}x\right)$

$\displaystyle =\frac{2}{L}\displaystyle\sum\limits_{n=1}^\infty b_n e^{\left( -k\frac{n^2\pi^2}{L^2}t\right)} \sin\left( \frac{n\pi}{L}x\right)$

 $b_n =$ $\displaystyle\int$ $\frac{L}{2}$$0$ $\hskip 1pt dx\hskip 35pt + \hskip 35pt$ $\displaystyle\int$ $L$$\frac{L}{2}$ $\hskip 1pt dx$

$\hskip 30pt$ u = dv = $\hskip 30pt$ u = dv =

$\hskip 30pt$ du = v = $\hskip 30pt$ du = v =

 $\hskip 12pt =$ $\Bigg\vert$ $\frac{L}{2}$$0$ $+\displaystyle\frac{L}{n\pi}$ $\displaystyle\int$ $\frac{L}{2}$$0$ $dx+$ $\Bigg\vert$ $L$$\frac{L}{2}$ $-\displaystyle\frac{L}{n\pi}$ $\displaystyle\int$ $L$$\frac{L}{2}$ $dx$

 $\hskip 12pt =$ $+$ $\Bigg\vert$ $\frac{L}{2}$$0$ $+$ $-$ $\Bigg\vert$ $L$$\frac{L}{2}$
 $\hskip 12pt =$

Your overall score for this problem is