6.2.7 \(u(-L)=0,u\left ( L\right ) =0\)

For eigenvalue

\[ \left \{\frac{\pi ^2}{4 L^2},\frac{\pi ^2}{L^2},\frac{9 \pi ^2}{4 L^2},\frac{4 \pi ^2}{L^2},\frac{25 \pi ^2}{4 L^2},\frac{9 \pi ^2}{L^2}\right \} \]

\[ \frac{\pi ^2 n^2}{4 L^2} \]

For eigenfunctions

\[ \left \{\sin \left (\frac{\pi (L+x)}{2 L}\right ),\sin \left (\frac{\pi (L+x)}{L}\right ),\sin \left (\frac{3 \pi (L+x)}{2 L}\right ),\sin \left (\frac{2 \pi (L+x)}{L}\right ),\sin \left (\frac{5 \pi (L+x)}{2 L}\right )\right \} \]

\[ \frac{1}{2} i \left (\left (-i e^{-\frac{i \pi x}{2 L}}\right )^n-\left (i e^{\frac{i \pi x}{2 L}}\right )^n\right ) \]