Internal problem ID [7859]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 279.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {\left (y^{2}+2 y+x \right ) y^{\prime }+\left (x +y\right )^{2} y^{2}+y \left (1+y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 120
dsolve((y(x)^2+2*y(x)+x)*diff(y(x),x)+(y(x)+x)^2*y(x)^2+y(x)*(y(x)+1)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {-c_{1} x +2 x^{2}-2+\sqrt {4 x^{4}-4 x^{3} c_{1}+x^{2} c_{1}^{2}-8 x^{2}+4 c_{1} x +16 x -8 c_{1}+4}}{2 c_{1}-4 x} \\ y \relax (x ) = -\frac {c_{1} x -2 x^{2}+\sqrt {4 x^{4}-4 x^{3} c_{1}+x^{2} c_{1}^{2}-8 x^{2}+4 c_{1} x +16 x -8 c_{1}+4}+2}{2 \left (c_{1}-2 x \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.108 (sec). Leaf size: 146
DSolve[(y[x]^2+2*y[x]+x)*y'[x]+(y[x]+x)^2*y[x]^2+y[x]*(y[x]+1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^2+\sqrt {\left (-x^2+c_1 x+1\right ){}^2+4 (x-c_1)}-c_1 x-1}{2 (x-c_1)} \\ y(x)\to \frac {-x^2+\sqrt {\left (-x^2+c_1 x+1\right ){}^2+4 (x-c_1)}+c_1 x+1}{2 (x-c_1)} \\ y(x)\to \frac {1}{2} \left (-\sqrt {x^2}-x\right ) \\ y(x)\to \frac {1}{2} \left (\sqrt {x^2}-x\right ) \\ \end{align*}