Question

Use a CAS to plot graphs to convince yourself that the equation $tan\alpha =-\alpha$ in Problem 38 has an infinite number of roots.

Explain why the negative roots of the equation can be ignored. Explain why$\lambda =0$is not an eigenvalue even though$\alpha =0$is an obvious solution of the equation $tan\alpha =-\alpha .$

Short answer

$y\left(x\right)=0$

Step-by-step solution

To find the boundary conditions

We see from the graph that $\tan x=-x$ has infinitely many roots. Since${\lambda }_n={\alpha }_n^2$, there are no new eigenvalues when ${\alpha }_n<0.$ For$\lambda =0$the differential equation $y^{\text{'}\text{'}}=0$ has general solution$y=c_{1}x+c_2.$The boundary conditions imply $c_1=c_2=0$so$y\left(x\right)=0$

Final proof

$y\left(x\right)=0$