Internal problem ID [8842]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1262.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x +1\right )^{2} y^{\prime \prime }+\left (x^{2}+x -1\right ) y^{\prime }-\left (x +2\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.054 (sec). Leaf size: 55
dsolve((x+1)^2*diff(diff(y(x),x),x)+(x^2+x-1)*diff(y(x),x)-(x+2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-x} \left (x +1\right ) \mathit {HD}\left (4, 4, -8, 12, \frac {x}{x +2}\right )+c_{2} \left (x +1\right ) \mathit {HD}\left (-4, 4, -8, 12, \frac {x}{x +2}\right ) {\mathrm e}^{\frac {x -1}{2 x +2}} \]
✓ Solution by Mathematica
Time used: 0.174 (sec). Leaf size: 42
DSolve[(-2 - x)*y[x] + (-1 + x + x^2)*y'[x] + (1 + x)^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} \left (c_2 \int _1^xe^{K[1]-\frac {1}{K[1]+1}} (K[1]+1)dK[1]+c_1\right ) \\ \end{align*}