Internal problem ID [9354]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1775.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x \left (x +1\right )^{2} y y^{\prime \prime }-x \left (x +1\right )^{2} \left (y^{\prime }\right )^{2}+2 \left (x +1\right )^{2} y y^{\prime }-a \left (x +2\right ) y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.082 (sec). Leaf size: 35
dsolve(x*(x+1)^2*y(x)*diff(diff(y(x),x),x)-x*(x+1)^2*diff(y(x),x)^2+2*(x+1)^2*y(x)*diff(y(x),x)-a*(x+2)*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \frac {\left (x +1\right )^{a} {\mathrm e}^{-a} {\mathrm e}^{\frac {c_{1}}{x}} {\mathrm e}^{-\frac {a}{x}}}{c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.127 (sec). Leaf size: 24
DSolve[-(a*(2 + x)*y[x]^2) + 2*(1 + x)^2*y[x]*y'[x] - x*(1 + x)^2*y'[x]^2 + x*(1 + x)^2*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 (x+1)^a e^{-\frac {a+c_1}{x}} \\ \end{align*}