Internal problem ID [8875]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1295.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {a \,x^{2} y^{\prime \prime }+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.055 (sec). Leaf size: 113
dsolve(a*x^2*diff(diff(y(x),x),x)+b*x*diff(y(x),x)+(c*x^2+d*x+f)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{-\frac {b}{2 a}} \WhittakerM \left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right )+c_{2} x^{-\frac {b}{2 a}} \WhittakerW \left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right ) \]
✓ Solution by Mathematica
Time used: 0.111 (sec). Leaf size: 214
DSolve[(f + d*x + c*x^2)*y[x] + b*x*y'[x] + a*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {i \sqrt {c} x}{\sqrt {a}}} x^{\frac {\sqrt {(a-b)^2-4 a f}+a-b}{2 a}} \left (c_1 \text {HypergeometricU}\left (\frac {\sqrt {(a-b)^2-4 a f}+\frac {i \sqrt {a} d}{\sqrt {c}}+a}{2 a},\frac {\sqrt {(a-b)^2-4 a f}+a}{a},\frac {2 i \sqrt {c} x}{\sqrt {a}}\right )+c_2 L_{-\frac {a+\frac {i d \sqrt {a}}{\sqrt {c}}+\sqrt {(a-b)^2-4 a f}}{2 a}}^{\frac {\sqrt {(a-b)^2-4 a f}}{a}}\left (\frac {2 i \sqrt {c} x}{\sqrt {a}}\right )\right ) \\ \end{align*}