Internal problem ID [8328]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 748.
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: 0.007 (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 \ln \relax (x ) c_{1}^{2}-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 \ln \relax (x ) c_{1}^{2}-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 \ln \relax (x ) c_{1}^{2}-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 \ln \relax (x ) c_{1}^{2}-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 \ln \relax (x ) c_{1}^{2}-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 \ln \relax (x ) c_{1}^{2}-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 \ln \relax (x ) c_{1}^{2}-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 \ln \relax (x ) c_{1}^{2}-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 \ln \relax (x ) c_{1}^{2}-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 \ln \relax (x ) c_{1}^{2}-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: 2.564 (sec). Leaf size: 287
DSolve[y'[x] == (y[x]*(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+\frac {1}{2} \sqrt {2916 x^2-864 (\log (x)+c_1){}^3}}}+\frac {\sqrt [3]{9 x+\frac {1}{6} \sqrt {2916 x^2-864 (\log (x)+c_1){}^3}}}{3^{2/3}} \\ y(x)\to \frac {\left (-1-i \sqrt {3}\right ) (\log (x)+c_1)}{\sqrt [3]{27 x+\frac {1}{2} \sqrt {2916 x^2-864 (\log (x)+c_1){}^3}}}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+c_1){}^3}}}{6 \sqrt [3]{2}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) (\log (x)+c_1)}{\sqrt [3]{27 x+\frac {1}{2} \sqrt {2916 x^2-864 (\log (x)+c_1){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+c_1){}^3}}}{6 \sqrt [3]{2}} \\ y(x)\to 0 \\ \end{align*}