I have the following two third order linear ODEs which have been arrived at after applying separation of variables to a coupled system of three PDEs. \begin{eqnarray} \lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \beta_h^2 F &=& 0,\\ V \lambda_c G''' - 2 V \lambda_c \beta_c G'' + \left( (\lambda_c \beta_c - 1) V \beta_c + \mu \right) G' + V \beta_c^2 G &=& 0, \end{eqnarray}

Here, $\lambda_h,\lambda_c,\beta_h,\beta_c$ are constants $>0$ while $\mu$ is the separation constant.$F$ is $F(x)$ while $G$ is $G(y)$. The b.c. are $F(0)=0,\frac{F''(0)}{F'(0)}=\beta_h,\frac{F''(1)}{F'(1)}=\beta_h$ while for $G$ they are $G(0)=0,\frac{G''(0)}{G'(0)}=\beta_c,\frac{G''(1)}{G'(1)}=\beta_c$.

So, i have two Eigenvalue BVP which need to be solved individually to attain the final solution as $\theta_w=e^{-\beta_hx}F'(x)e^{-\beta_cy}G'(y)$.


I tried to find the eigen value for the ODE in $F$ with the constants as $\lambda_h=0.02,\beta_h=10$ in Mathematica using a package called CompoundMatrixMethod by Simon Pearce and it gave me the following resultsThe intersecting points on x are eigenvalues

Similarly, for the ODE in $G$ with the constants as $\lambda_c=0.02,\beta_c=10, V=1$, the results wereenter image description here

The intersecting points with the x-axis are the eigenvalues.

It is quite clear that for $F$ there are possibly infinite negative eigenvalues and $G$ has possibly infinite positive EVs.

I had two questions:

1. Can someone try (or is it even possible ) to arrive at an analytic pattern to how these eigenvalues repeat ?

2. Does the increasing oscillatory nature of the EVs as they move further away from the origin signal that a analytic solution cannot be reached ?

I am curious about the pattern, because a lot of PDE books show that after variable separation the ODEs have a series of eigen values following a trend, which is then used to generate the eigen functions and individual general solutions.


Out of curiosity i just plotted $e^{0.07x}sin(x)$ and found it to be like this. This seemed quite similar to the $G$ case.enter image description here


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