Question

Use a root-finding application of a CAS to approximate the first four eigenvalues ${\lambda }_1,{\lambda }_2,{\lambda }_3,$and${\lambda }_4$ for the BVP in Problem 38 .

Short answer

$$\begin{gathered} {\lambda }_1=4.11586,y_1\left(x\right)=sin(2.02876x) \\ {\lambda }_2=24.1393,y_2\left(x\right)=sin(4.91318x) \\ {\lambda }_3=63.6591,y_3\left(x\right)=sin(7.97867x) \\ {\lambda }_4=122.889,y_4\left(x\right)=sin(11.0855x) \end{gathered}$$

Step-by-step solution

To Find the root finding of a CAS

Using a CAS we find that the first four nonnegative roots of are approximately $2.02876,4.91318,7.97867,$and $11.0855.$

The corresponding eigenvalues are$4.11586,24.1393,63.6591,$ and $122.889$with eigenfunctions $\sin (2.02876x),\sin (4.91318x),\sin (7.97867x)$and $\sin (11.0855x).$

Final proof

$$\begin{gathered} {\lambda }_1=4.11586,y_1\left(x\right)=sin(2.02876x) \\ {\lambda }_2=24.1393,y_2\left(x\right)=sin(4.91318x) \\ {\lambda }_3=63.6591,y_3\left(x\right)=sin(7.97867x) \\ {\lambda }_4=122.889,y_4\left(x\right)=sin(11.0855x) \end{gathered}$$