# Evaluating Eigen values of [Cubic ODE]

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 ?