How to choose Eigenvaues for system extending in perpendicular direction?

I have two linear third order ODEs with a separation constant (eigenvalue parameter) on a rectangular domain where $$x \in (0,1)$$ and $$y \in (0,1)$$as follows:

$$\lambda_h F''' - 2 \lambda_h \beta_h F'' + ( (\lambda_h \beta_h - 1) \beta_h ) F' + \beta_h^2 F = \mu F',\\ V \lambda_c G''' - 2 V \lambda_c \beta_c G'' +( (\lambda_c \beta_c - 1) V \beta_c ) G' + V \beta_c^2 G = -\mu G',$$ $$\mu \in \mathbb{R}$$.

The b.c for $$F(x)$$ are $$F(0) = 0, \frac{F''(0)}{F'(0)}=\beta_h,\frac{F''(1)}{F'(1)}=\beta_h$$

For $$G(y)$$ they are $$G(0) = 0, \frac{G''(0)}{G'(0)}=\beta_c,\frac{G''(1)}{G'(1)}=\beta_c$$

For these bc(s) and $$\lambda_h=0.02, \beta_h = 10$$, I calculated the Eigenvalues using chebfun in MATLAB which came out to be

F = -37.6413, -32.9463, -28.6002, -24.5873, -20.8885, -17.4846, -14.3643 -11.5367, -9.0383, -6.9287,...... 

G = 6.9287, 9.0383, 11.5367, 14.3643, 17.4846, 20.8885, 24.5873, 28.6002, 32.9463, 37.6413, .....

I now need to find the solutions to these two separated ODEs, to form my final solution.

What bothers me is that EVs of $$F$$ are in infinite numbers in the $$-x$$ direction, while of $$G$$ are in infinite numbers in $$+y$$ direction with the same magnitudes.

I cannot figure out what EVs should I choose and how many i should to build my solution ? I know that they are separated by the same constant. Basically, how should i proceed. haven't gotten any clue from texts.

Steps after finding the EVs

The general solution will be of the form

$$F(x) = \sum_k C_k e^{-\delta_k(\mu)x}$$

where $$\delta_k(\mu)$$ is a root of the characteristic equation dependent on $$\mu$$. There would be three roots of the char. eqn. So each EV when substituted in the characteristic equation should give three roots which would then be used to calculate $$C_1,C_2,C_3$$ constants using the b.c.

This is where i am facing problems in determining, how many and which EVs should be considered.

• A remark : is $V$ (which is factorizable in the LHS of the second equation) a known or unknown quantity ? This question is linked to the fact that I think that replacing $\mu$ by $\mu/V$ would may be simplify this issue (with Neumann boundary conditions, more difficult to tackle than Cauchy initial conditions). – Jean Marie Jan 22 at 22:01
• What are the values of $\lambda_c$ and $\beta_c$ ? – Jean Marie Jan 22 at 22:16
• @JeanMarie $\lambda_c=0.02$ $\beta_c=10$ and $V=1$ can be consodered a case – Indrasis Mitra Jan 22 at 23:52
• @JeanMarie I tired the problem with $V=1$, all the eigenvalues that i have reported for $G(y)$ are calculated for $V=1$. I am just at a loss in interpreting them. I have edited my original question to reflect how i am trying to approach the problem. – Indrasis Mitra Jan 23 at 2:24
• Now posted to MO, mathoverflow.net/questions/321528/… – Gerry Myerson Jan 23 at 11:59