Internal problem ID [8770]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1190.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+y^{\prime } x^{2}+\left (a x +b \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 41
dsolve(x^2*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+(a*x+b)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {x}{2}} \WhittakerM \left (a , \frac {\sqrt {1-4 b}}{2}, x\right )+c_{2} {\mathrm e}^{-\frac {x}{2}} \WhittakerW \left (a , \frac {\sqrt {1-4 b}}{2}, x\right ) \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 95
DSolve[(b + a*x)*y[x] + x^2*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} x^{\frac {1}{2} \left (\sqrt {1-4 b}+1\right )} \left (c_1 \text {HypergeometricU}\left (\frac {1}{2} \left (-2 a+\sqrt {1-4 b}+1\right ),\sqrt {1-4 b}+1,x\right )+c_2 L_{a-\frac {1}{2} \sqrt {1-4 b}-\frac {1}{2}}^{\sqrt {1-4 b}}(x)\right ) \\ \end{align*}