6.2.1 \(u(0)=0,u(L)=0\)

For eigenvalue

\[ \left \{\frac{\pi ^2}{L^2},\frac{4 \pi ^2}{L^2},\frac{9 \pi ^2}{L^2},\frac{16 \pi ^2}{L^2},\frac{25 \pi ^2}{L^2}\right \} \]

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

For eigenfunctions

\[ \left \{\sin \left (\frac{\pi x}{L}\right ),\sin \left (\frac{2 \pi x}{L}\right ),\sin \left (\frac{3 \pi x}{L}\right ),\sin \left (\frac{4 \pi x}{L}\right ),\sin \left (\frac{5 \pi x}{L}\right )\right \} \]

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