The twist angle $\theta\left(x,t\right)$ of a torsionally vibrating shaft of unit length is determined from: $$\displaystyle a^2\frac{\partial^2 \theta}{\partial x^2}=\frac{\partial^2 \theta}{\partial t^2}, 0< x < 1, t>0$$ $$\displaystyle \theta\left(0,t\right) = 0,\hskip 20pt \frac{\partial \theta}{\partial x}\Bigg\vert_{x=1}=0, t>0$$ $$\displaystyle \theta\left(x,0\right) = -3x^{2}+x-3, \hskip 10pt\frac{\partial \theta}{\partial t}\Bigg\vert_{t=0}=0, t>0$$
Then $\displaystyle\theta\left(x,t\right) = \sum\limits_{n=1}^\infty A_n\cos\!\left(a\frac{2n-1}{2}\pi t\right)\sin\!\left(\frac{2n-1}{2}\pi x\right)$
where

$A_n =$

$\hskip 10pt\int\limits_0^1$dx ------------------------------------------------------------- $\hskip 10pt\int\limits_0^1$dx

$\hskip 15pt =$