Let $\alpha_{i},\beta_{i} \in \mathbb{C}$, $\forall i= 1,2$ and $0 < a < b < \infty$. \begin{align} &f^{\prime \prime} + \alpha_{1} f = \alpha_{2} g^{\prime} \quad \text{in} \quad [a,b] \\ &g^{\prime \prime} - \beta_{1} g = \beta_{2}f^{\prime} \quad \text{in} \quad [a,b]. \end{align} with $f(b) = g(a)=g(b)=0$.
Well, doing the appropriate manipulations we arrive at a problem of analyzing the roots of a biquadratic equation and transforming it into a problem of determining the roots of a quadratic equation - I have no doubt about this part - so I'll post the result straight away. I get the following solutions: \begin{align} f(x) = \sum_{j=1}^{4}c_{j}e^{r_{j}x} \quad \text{and} \quad g(x) = \sum_{j=1}^{4}d_{j}e^{s_{j}x} \end{align} with $r_{1},,s_{j} \in \mathbb{C}$. I'm having difficulty finishing these constants $c_{j}$ and $d_{j}$. I've tried substituting these equations into the system and using boundary conditions, but I'm not getting anything relevant. How to proceed intelligently?
