I wish to integrate the following differential equation $$\ddot{u}(x)=(a+bx)u(x)$$ to solve for $u(x)$. I tried to characterize it as a Sturm-Liouville problem and introduce an integrating factor, but I was unable to make much headway. I would like to try integrating it without appealing to a power series and recurrence relationship. (I was already capable of producing a power series solution, but it did not elucidate me.) If the differential equation was first order, I would have immediately used separability.
I also tried the ansatz $u(x) = e^{\pm \sqrt a x} w(x)$ which produced the equation $$ \ddot w(x) + 2\sqrt {a} \dot w(x) - b x w(x) = 0 $$
Put $v(x)=u(cx-d)$ in $\ddot{v}=xv$ and solve to get $v(x)=u\big(\frac{x}{\sqrt[3]{a}}-\frac{b}{a}\big)$, or $u(x)=v\big(\sqrt[3]{a}(x+\frac{b}{a})\big)$.
The solutions for $v(x)$ are spanned by the Airy functions $\mathrm{Ai}(x)$ and $\mathrm{Bi}(x)$, by definition.