Internal problem ID [9212]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 877.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class C`]]
\[ \boxed {y^{\prime }-\frac {-2 y x +2 x^{3}-2 x -y^{3}+3 y^{2} x^{2}-3 x^{4} y+x^{6}}{x^{2}-y-1}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 71
dsolve(diff(y(x),x) = (-2*x*y(x)+2*x^3-2*x-y(x)^3+3*x^2*y(x)^2-3*y(x)*x^4+x^6)/(-y(x)+x^2-1),y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {-2 x^{2} c_{1} +2 x^{3}+\sqrt {2 c_{1} -2 x +1}-1}{2 \left (-x +c_{1} \right )} y \left (x \right ) = \frac {2 x^{2} c_{1} -2 x^{3}+\sqrt {2 c_{1} -2 x +1}+1}{-2 x +2 c_{1}} \end{align*}
✓ Solution by Mathematica
Time used: 0.35 (sec). Leaf size: 54
DSolve[y'[x] == (-2*x + 2*x^3 + x^6 - 2*x*y[x] - 3*x^4*y[x] + 3*x^2*y[x]^2 - y[x]^3)/(-1 + x^2 - y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^2+\frac {1}{-1+\sqrt {-2 x+c_1}} y(x)\to x^2-\frac {1}{1+\sqrt {-2 x+c_1}} y(x)\to x^2 \end{align*}