I'm trying to solve a particular differential equation:
$$(c*z+d)*y''(z)+c*y'(z)=(a*z+b)*y(z)$$
wherea, b, c and d are constant and y is a function of z.
This is a particular Sturm-Liouville differential equation (= Kummer's equation apparently). I find a perfectly suitable solution in terms of confluent hypergeometric function of the first and second kind when $c$ and $a$ are $>0$ (thanks wolframalpha!).
Unfortunately whein either $c$ or $a$ are negative, ro real solution appears because somewhere in the solution I always have $\sqrt{a}$ or $c^{\frac{3}{2}}$.
Is it possible that no real solutions exist for this particular differential equation ?