PROBLEM
Consider the problem \begin{eqnarray} u_t && = c^2 u_{xx}, 0< x< \pi, t>0,\\ u(0,t) && = 0= u_x(\pi,t) + \gamma u(\pi,t), 0 \le t,\\ u(x,0) &&= f(x), 0 \le x \le \pi, \end{eqnarray} where $\gamma \in R$ and $f$ is a given real function. $$
My problem is: deduct grafically that the equation bellow is satisfy for a infinity and countable of positive values of $\lambda$ for the equation
$$
\sqrt{\lambda}\cos (\sqrt{\lambda} \pi) + \gamma \sin(\sqrt{\lambda} \pi) =0
$$
and, if $\lambda_1< \lambda_2 < ... < \lambda_n<...$ are eingenvalues for the ODE, obtained through the method of separation of variables $u(x,t) = \varphi(x) \psi(t)$, i.e
\begin{eqnarray}
-\varphi '' && = \lambda \varphi,\\
\varphi(0) && = 0 = \varphi '(\pi) + \gamma \varphi(\pi).
\end{eqnarray}
than $\lambda_n \to +\infty$, when $n \to + \infty$. Write the solution $u_n(x,t)$ which corresponds to the eigenvalue $\lambda_n$
MY ATTEMPT
I did a graphical analysis considering that the equation is are level curves of the function $$ \xi (\lambda, \gamma) = \sqrt{\lambda}\cos (\sqrt{\lambda} \pi) + \gamma \sin(\sqrt{\lambda} \pi) $$ The "roots" of the $\xi (\lambda, \gamma) = 0$ are the points where the curves cross the $\lambda$'s axis.
Note that, by the graph, it is possible to identify that the values of the lambdas are strictly increasing and that they increase more and more.
But I don't know if my analysis is correct, or if I'm being misled by my geometric intuition. Also, I don't know how this will help me find the $\lambda_n$ eigenvalue to complete the last part of the exercise. I tried in every way to solve this equation "analytically". But I don't get any results. Any idea??
Thanks!!