Internal problem ID [8468]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 888.
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)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {6 x^{2} y-2 x +1-5 x^{3} y^{2}-2 x y+y^{3} x^{4}}{x^{2} \left (x^{2} y-x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 79
dsolve(diff(y(x),x) = 1/x^2*(6*x^2*y(x)-2*x+1-5*x^3*y(x)^2-2*x*y(x)+y(x)^3*x^4)/(x^2*y(x)-x+1),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {\frac {c_{1} x +2}{x}}\, x -x +1}{x^{2} \left (\sqrt {\frac {c_{1} x +2}{x}}-1\right )} \\ y \relax (x ) = \frac {\sqrt {\frac {c_{1} x +2}{x}}\, x +x -1}{x^{2} \left (\sqrt {\frac {c_{1} x +2}{x}}+1\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.455 (sec). Leaf size: 73
DSolve[y'[x] == (1 - 2*x - 2*x*y[x] + 6*x^2*y[x] - 5*x^3*y[x]^2 + x^4*y[x]^3)/(x^2*(1 - x + x^2*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(-1+c_1) x+\sqrt {\frac {2}{x}+c_1}+3}{x (2+(-1+c_1) x)} \\ y(x)\to \frac {x+\frac {1}{1+\frac {1}{\sqrt {\frac {2}{x}+c_1}}}-1}{x^2} \\ y(x)\to \frac {1}{x} \\ \end{align*}