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I have a DE (resulting from variable separation applied on a PDE with $\mu$ acting as the separation coefficient, all other terms are constant and $>0$)

$\lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \beta_h^2 F=0$

with boundary conditions: $$F(0)=0$$ $$\frac{F''(0)}{F'(0)}=\beta_h$$ $$\frac{F''(1)}{F'(1)}=\beta_h$$ This needs to be solved for its eigen values so i need to form a characteristic equation and solve for its roots. The problem is that it turns out to be cubic and since all the constants are unknowns, i have problems in evaluating cases of $\mu>0$, $\mu<0$ and $\mu=0$ which is pretty easy for a second order ODE.

Is there any different way of approaching this problem ? Can Laplace transform be a way ?

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