Internal problem ID [8918]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1339.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (a \left (2+b \right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (a x +1\right ) x^{2}}+\frac {\left (a b x -c d \right ) y}{\left (a x +1\right ) x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 89
dsolve(diff(diff(y(x),x),x) = -(a*(b+2)*x^2+(c-d+1)*x)/(a*x+1)/x^2*diff(y(x),x)-(a*b*x-c*d)/(a*x+1)/x^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{d} \left (a x +1\right )^{-b +c -d} \hypergeom \left (\left [c , 1-b +c \right ], \left [1+d +c \right ], -a x \right )+c_{2} x^{-c} \left (a x +1\right )^{-b +c -d} \hypergeom \left (\left [-d , 1-b -d \right ], \left [1-d -c \right ], -a x \right ) \]
✓ Solution by Mathematica
Time used: 0.129 (sec). Leaf size: 66
DSolve[y''[x] == -(((-(c*d) + a*b*x)*y[x])/(x^2*(1 + a*x))) - (((1 + c - d)*x + a*(2 + b)*x^2)*y'[x])/(x^2*(1 + a*x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 a^{-c} x^{-c} \, _2F_1(1-c,b-c;-c-d+1;-a x)+c_2 a^d x^d \, _2F_1(d+1,b+d;c+d+1;-a x) \\ \end{align*}