This is essentially problem #3 on page 178.
Consider the SLP
$$-y''=\lambda y, \: y(0)+y'(0)=0, y(1) = 0.$$
Find an equation to describe the the eigenvalues. Use this to graphically explain why there are infinitely many and discuss their rate of growth.
Comments
Solving the differential equation $y'' = -\lambda y$,
$$y = a\cos\left(\sqrt{\lambda}x\right) + b\sin\left(\sqrt{\lambda}x\right)$$
and
$$y' = -a\sqrt{\lambda}\sin\left(\sqrt{\lambda}x\right) + b\sqrt{\lambda}\cos\left(\sqrt{\lambda}x\right).$$
From our initial conditions,
$$y(0) + y'(0) = a + b\sqrt{\lambda} = 0,$$
so
$$a = -b\sqrt{\lambda}.$$
We also know
$$y(1) = a\cos\left(\sqrt{\lambda}\right) + b\sin\left(\sqrt{\lambda}\right) = -b\sqrt{\lambda}\cos\left(\sqrt{\lambda}\right) + b\sin\left(\sqrt{\lambda}\right) = 0$$
$$=> b\sqrt{\lambda}\cos\left(\sqrt{\lambda}\right) = b\sin\left(\sqrt{\lambda}\right),$$
so
$$\sqrt{\lambda} = \tan\left(\sqrt{\lambda}\right).$$
Hence, our eigenvalues are the points where $\sqrt{\lambda}$ and $\tan\left(\sqrt{\lambda}\right)$ intersect. By graphing these functions, one can see that there are infinitely many such intersections. One can also observe that the distance between each intersection grows as $\lambda$ increases, so $\lambda_{n+1} - \lambda_n$ increases with $n.$