Internal problem ID [8799]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1219.
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}+2 x f \relax (x ) y^{\prime }+\left (f^{\prime }\relax (x ) x +f \relax (x )^{2}-f \relax (x )+a \,x^{2}+x b +c \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 79
dsolve(x^2*diff(diff(y(x),x),x)+2*x*f(x)*diff(y(x),x)+(x*diff(f(x),x)+f(x)^2-f(x)+a*x^2+b*x+c)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \WhittakerM \left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i \sqrt {a}\, x \right ) {\mathrm e}^{-\left (\int \frac {f \relax (x )}{x}d x \right )}+c_{2} \WhittakerW \left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i \sqrt {a}\, x \right ) {\mathrm e}^{-\left (\int \frac {f \relax (x )}{x}d x \right )} \]
✓ Solution by Mathematica
Time used: 0.079 (sec). Leaf size: 151
DSolve[y[x]*(c + b*x + a*x^2 - f[x] + f[x]^2 + x*Derivative[1][f][x]) + 2*x*f[x]*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (c_1 \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {i b}{\sqrt {a}}+\sqrt {1-4 c}+1\right ),\sqrt {1-4 c}+1,2 i \sqrt {a} x\right )+c_2 L_{\frac {1}{2} \left (-\frac {i b}{\sqrt {a}}-\sqrt {1-4 c}-1\right )}^{\sqrt {1-4 c}}\left (2 i \sqrt {a} x\right )\right ) \exp \left (\int _1^x\frac {-2 f(K[1])-2 i \sqrt {a} K[1]+\sqrt {1-4 c}+1}{2 K[1]}dK[1]\right ) \\ \end{align*}