Internal problem ID [9064]
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]
\[ \boxed {y^{\prime }-\frac {y \left (-y+x \right )}{x \left (x -y^{3}\right )}=0} \]
✓ 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 \left (x \right ) = \frac {\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 \ln \left (x \right )^{2} c_{1} -72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{3}-\frac {3 \left (-\frac {2 \ln \left (x \right )}{3}+\frac {2 c_{1}}{3}\right )}{\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 \ln \left (x \right )^{2} c_{1} -72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}} y \left (x \right ) = -\frac {\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 \ln \left (x \right )^{2} c_{1} -72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {-\ln \left (x \right )+c_{1}}{\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 \ln \left (x \right )^{2} c_{1} -72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 \ln \left (x \right )^{2} c_{1} -72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{3}+\frac {-2 \ln \left (x \right )+2 c_{1}}{\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 \ln \left (x \right )^{2} c_{1} -72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} y \left (x \right ) = -\frac {\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 \ln \left (x \right )^{2} c_{1} -72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {-\ln \left (x \right )+c_{1}}{\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 \ln \left (x \right )^{2} c_{1} -72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 \ln \left (x \right )^{2} c_{1} -72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}{3}+\frac {-2 \ln \left (x \right )+2 c_{1}}{\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 \ln \left (x \right )^{2} c_{1} -72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 5.771 (sec). Leaf size: 320
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 \sqrt [3]{2} (-\log (x)+c_1)}{\sqrt [3]{54 x+2 \sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}}}-\frac {\sqrt [3]{54 x+2 \sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}}}{3 \sqrt [3]{2}} y(x)\to \frac {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) (-\log (x)+c_1)}{\sqrt [3]{54 x+2 \sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{54 x+2 \sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}}}{6 \sqrt [3]{2}} y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{54 x+2 \sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}}}{6 \sqrt [3]{2}}-\frac {i \sqrt [3]{2} \left (\sqrt {3}-i\right ) (-\log (x)+c_1)}{\sqrt [3]{54 x+2 \sqrt {729 x^2+(-6 \log (x)+6 c_1){}^3}}} y(x)\to 0 \end{align*}