The following ordinary differential equation for $F(y)$ is given. $$ (\alpha + \beta A y)\left(\frac{\partial^2 F}{\partial y^2} + \beta^2 F\right) = 0 , $$ The boundary conditions are $F(y \to \pm \infty) = 0$.
At first, I assumed that it is possible to simply omit the term $\alpha + \beta A y$, even though it might be zero for $y = -\alpha/(\beta A)$. This would result in: $$ F(y) = C_1 \sin(\beta \, y) + C_2 \cos(\beta \, y) $$ However this only satisfies the boundary conditions if $C_1 = C_2 = 0$.
- Does this mean that no solution fulfilling the given BCs exists?
- What would be the correct approach to handle this type of problem?