Internal problem ID [7816]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 236.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {x \left (y+4\right ) y^{\prime }-y^{2}-2 y-2 x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 145
dsolve(x*(y(x)+4)*diff(y(x),x)-y(x)^2-2*y(x)-2*x=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (4+x \right )^{\frac {3}{2}} \sqrt {\frac {c_{1} x +4 c_{1}-4}{4+x}}\, x +4 x^{\frac {3}{2}}+16 \sqrt {x}}{\left (4+x \right )^{\frac {3}{2}} \sqrt {\frac {c_{1} x +4 c_{1}-4}{4+x}}-x^{\frac {3}{2}}-4 \sqrt {x}} \\ y \relax (x ) = \frac {\left (4+x \right )^{\frac {3}{2}} \sqrt {\frac {c_{1} x +4 c_{1}-4}{4+x}}\, x -4 x^{\frac {3}{2}}-16 \sqrt {x}}{\left (4+x \right )^{\frac {3}{2}} \sqrt {\frac {c_{1} x +4 c_{1}-4}{4+x}}+x^{\frac {3}{2}}+4 \sqrt {x}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.005 (sec). Leaf size: 90
DSolve[x*(y[x]+4)*y'[x]-y[x]^2-2*y[x]-2*x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 x+\sqrt {x} \sqrt {x+4} \sqrt {-\frac {4}{x+4}+c_1}}{-1+c_1} \\ y(x)\to -4+\frac {1}{\frac {1}{x+4}+\frac {\sqrt {x}}{(x+4)^{3/2} \sqrt {-\frac {4}{x+4}+c_1}}} \\ y(x)\to x \\ \end{align*}