Boundary Conditions of the Third Kind

[Dustinsfl]

\begin{align}
\varphi''+\lambda\varphi &= 0, & \quad 0< x < L\\
\varphi'(0) &= 0 &\\
\varphi'(L)+h\varphi(L) &=0, & \quad h\in\mathbb{R}
\end{align}
$$
\varphi = A\cos x\sqrt{\lambda} + B\frac{\sin x\sqrt{\lambda}}{\sqrt{\lambda}}
$$
Since $\varphi'(0) = 0$, $\varphi = A\cos x\sqrt{\lambda}$. Then
$$
-\sqrt{\lambda}\sin L\sqrt{\lambda}+h\cos L\sqrt{\lambda} = 0 \Leftrightarrow \sqrt{\lambda}\tan L\sqrt{\lambda} = h
$$
Multiplying be $L$.
$$
L\sqrt{\lambda}\tan L\sqrt{\lambda} = Lh
$$
Let $L\sqrt{\lambda} = s$. Then $s\tan s = Lh$. By definition, $\tan (-s) = -\tan s$ and $\lambda = \left(\dfrac{s}{L}\right)^2 = \left(\dfrac{-s}{L}\right)^2$.
\begin{align}
y_1 &= \tan s, & \quad 0\leq s <\infty\\
y_2 &= \frac{Lh}{s}, & \quad 0\leq s <\infty
\end{align}

Why does $Lh = \dfrac{\pi}{2}\mbox{?}$

 

[Ackbach]

\begin{align}
\varphi''+\lambda\varphi &= 0, & \quad 0< x < L\\
\varphi'(0) &= 0 &\\
\varphi'(L)+h\varphi(L) &=0, & \quad h\in\mathbb{R}
\end{align}
$$
\varphi = A\cos x\sqrt{\lambda} + B\frac{\sin x\sqrt{\lambda}}{\sqrt{\lambda}}
$$
Since $\varphi'(0) = 0$, $\varphi = A\cos x\sqrt{\lambda}$. Then
$$
-\sqrt{\lambda}\sin L\sqrt{\lambda}+h\cos L\sqrt{\lambda} = 0 \Leftrightarrow \sqrt{\lambda}\tan L\sqrt{\lambda} = h
$$
Multiplying be $L$.
$$
L\sqrt{\lambda}\tan L\sqrt{\lambda} = Lh
$$
Let $L\sqrt{\lambda} = s$. Then $s\tan s = Lh$. By definition, $\tan (-s) = -\tan s$ and $\lambda = \left(\dfrac{s}{L}\right)^2 = \left(\dfrac{-s}{L}\right)^2$.
\begin{align}
y_1 &= \tan s, & \quad 0\leq s <\infty\\
y_2 &= \frac{Lh}{s}, & \quad 0\leq s <\infty
\end{align}

Why does $Lh = \dfrac{\pi}{2}\mbox{?}$

I don't think it does. From this plot, you can see that the solutions are going to get closer and closer to integer multiples of $\pi$, not $\pi/2$.