Internal problem ID [8509]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 931.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {y^{\prime }-\frac {-3 y x^{2}-2 x^{3}-2 x -x y^{2}-y+x^{3} y^{3}+3 x^{4} y^{2}+3 x^{5} y+x^{6}}{x \left (y x +x^{2}+1\right )}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 73
dsolve(diff(y(x),x) = (-3*x^2*y(x)-2*x^3-2*x-x*y(x)^2-y(x)+x^3*y(x)^3+3*x^4*y(x)^2+3*x^5*y(x)+x^6)/x/(x*y(x)+x^2+1),y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {\sqrt {-2 x +c_{1}}\, x^{2}-x^{2}-1}{x \left (\sqrt {-2 x +c_{1}}-1\right )} \\ y \left (x \right ) = -\frac {\sqrt {-2 x +c_{1}}\, x^{2}+x^{2}+1}{x \left (\sqrt {-2 x +c_{1}}+1\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.405 (sec). Leaf size: 60
DSolve[y'[x] == (-2*x - 2*x^3 + x^6 - y[x] - 3*x^2*y[x] + 3*x^5*y[x] - x*y[x]^2 + 3*x^4*y[x]^2 + x^3*y[x]^3)/(x*(1 + x^2 + x*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x+\frac {1}{-x+x \sqrt {-2 x+c_1}} \\ y(x)\to -x-\frac {1}{x+x \sqrt {-2 x+c_1}} \\ y(x)\to -x \\ \end{align*}