6.2.4 \(u(0)=0,u^{\prime }(L)=0\)

Eigenvalue are the same as above

\[ \left \{\frac{\pi ^2}{4 L^2},\frac{9 \pi ^2}{4 L^2},\frac{25 \pi ^2}{4 L^2},\frac{49 \pi ^2}{4 L^2},\frac{81 \pi ^2}{4 L^2},\frac{121 \pi ^2}{4 L^2}\right \} \]

\[ \frac{\pi ^2 (1-2 n)^2}{4 L^2} \]

For eigenfunctions

\[ \left \{\sin \left (\frac{\pi x}{2 L}\right ),\sin \left (\frac{3 \pi x}{2 L}\right ),\sin \left (\frac{5 \pi x}{2 L}\right ),\sin \left (\frac{7 \pi x}{2 L}\right ),\sin \left (\frac{9 \pi x}{2 L}\right ),\sin \left (\frac{11 \pi x}{2 L}\right )\right \} \]

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