Internal problem ID [8309]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 729.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y \left (x -y\right )}{x \left (x -y^{3}\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 497
dsolve(diff(y(x),x) = y(x)*(x-y(x))/x/(x-y(x)^3),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-27 x +3 \sqrt {24 c_{1}^{3}-72 c_{1}^{2} \ln \relax (x )+72 \ln \relax (x )^{2} c_{1}-24 \ln \relax (x )^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{3}-\frac {3 \left (\frac {2 c_{1}}{3}-\frac {2 \ln \relax (x )}{3}\right )}{\left (-27 x +3 \sqrt {24 c_{1}^{3}-72 c_{1}^{2} \ln \relax (x )+72 \ln \relax (x )^{2} c_{1}-24 \ln \relax (x )^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-27 x +3 \sqrt {24 c_{1}^{3}-72 c_{1}^{2} \ln \relax (x )+72 \ln \relax (x )^{2} c_{1}-24 \ln \relax (x )^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {c_{1}-\ln \relax (x )}{\left (-27 x +3 \sqrt {24 c_{1}^{3}-72 c_{1}^{2} \ln \relax (x )+72 \ln \relax (x )^{2} c_{1}-24 \ln \relax (x )^{3}+81 x^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-27 x +3 \sqrt {24 c_{1}^{3}-72 c_{1}^{2} \ln \relax (x )+72 \ln \relax (x )^{2} c_{1}-24 \ln \relax (x )^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{3}+\frac {2 c_{1}-2 \ln \relax (x )}{\left (-27 x +3 \sqrt {24 c_{1}^{3}-72 c_{1}^{2} \ln \relax (x )+72 \ln \relax (x )^{2} c_{1}-24 \ln \relax (x )^{3}+81 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-27 x +3 \sqrt {24 c_{1}^{3}-72 c_{1}^{2} \ln \relax (x )+72 \ln \relax (x )^{2} c_{1}-24 \ln \relax (x )^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {c_{1}-\ln \relax (x )}{\left (-27 x +3 \sqrt {24 c_{1}^{3}-72 c_{1}^{2} \ln \relax (x )+72 \ln \relax (x )^{2} c_{1}-24 \ln \relax (x )^{3}+81 x^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-27 x +3 \sqrt {24 c_{1}^{3}-72 c_{1}^{2} \ln \relax (x )+72 \ln \relax (x )^{2} c_{1}-24 \ln \relax (x )^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{3}+\frac {2 c_{1}-2 \ln \relax (x )}{\left (-27 x +3 \sqrt {24 c_{1}^{3}-72 c_{1}^{2} \ln \relax (x )+72 \ln \relax (x )^{2} c_{1}-24 \ln \relax (x )^{3}+81 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.886 (sec). Leaf size: 278
DSolve[y'[x] == ((x - y[x])*y[x])/(x*(x - y[x]^3)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 (-\log (x)+c_1)}{\sqrt [3]{27 x+\sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}}}-\frac {1}{3} \sqrt [3]{27 x+\sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) (-\log (x)+c_1)}{\sqrt [3]{27 x+\sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}}}+\frac {1}{6} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 x+\sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}} \\ y(x)\to \frac {1}{6} \left (1-i \sqrt {3}\right ) \sqrt [3]{27 x+\sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}}-\frac {i \left (\sqrt {3}-i\right ) (-\log (x)+c_1)}{\sqrt [3]{27 x+\sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}}} \\ y(x)\to 0 \\ \end{align*}