I would like to know if the following differential equation is solvable and what kind of steps I need to take or where to look to learn the steps. Sadly i do not know enough about differential equations to solve it.
I think i do not want to write it al out by hand a solver(function?) with Matlab would be sufficient. But i do not know if a solution is possible at all.
$$ \frac{d^4 v}{d x^4} = -c(x)v(x) $$
Where $ c(x) $ is for instance: $$ c(x) = 3x$$
With boundary conditions: $$ v'''(0) = -F$$ $$ v'(0)= 0$$
$$ v'''(L) = F$$ $$ v'(L)= 0$$