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

For eigenvalue

\[ \left \{0,\frac{\pi ^2}{L^2},\frac{4 \pi ^2}{L^2},\frac{9 \pi ^2}{L^2},\frac{16 \pi ^2}{L^2}\right \} \]

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

For eigenfunctions

\[ \left \{1,\cos \left (\frac{\pi x}{L}\right ),\cos \left (\frac{2 \pi x}{L}\right ),\cos \left (\frac{3 \pi x}{L}\right ),\cos \left (\frac{4 \pi x}{L}\right )\right \} \]

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