Internal problem ID [9083]
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]
\[ \boxed {y^{\prime }-\frac {y \left (x +y\right )}{x \left (x +y^{3}\right )}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 316
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 c_{1}^{3}-72 \ln \left (x \right ) c_{1}^{2}-72 \ln \left (x \right )^{2} c_{1} -24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{\frac {2}{3}}+6 \ln \left (x \right )+6 c_{1}}{3 \left (27 x +3 \sqrt {-24 c_{1}^{3}-72 \ln \left (x \right ) c_{1}^{2}-72 \ln \left (x \right )^{2} c_{1} -24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (-i \sqrt {3}-1\right ) \left (27 x +3 \sqrt {-24 c_{1}^{3}-72 \ln \left (x \right ) c_{1}^{2}-72 \ln \left (x \right )^{2} c_{1} -24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{\frac {2}{3}}}{6}+\left (i \sqrt {3}-1\right ) \left (\ln \left (x \right )+c_{1} \right )}{\left (27 x +3 \sqrt {-24 c_{1}^{3}-72 \ln \left (x \right ) c_{1}^{2}-72 \ln \left (x \right )^{2} c_{1} -24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (27 x +3 \sqrt {-24 c_{1}^{3}-72 \ln \left (x \right ) c_{1}^{2}-72 \ln \left (x \right )^{2} c_{1} -24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{\frac {2}{3}}}{6}+\left (-i \sqrt {3}-1\right ) \left (\ln \left (x \right )+c_{1} \right )}{\left (27 x +3 \sqrt {-24 c_{1}^{3}-72 \ln \left (x \right ) c_{1}^{2}-72 \ln \left (x \right )^{2} c_{1} -24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.664 (sec). Leaf size: 291
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 \sqrt [3]{2} (\log (x)+c_1)}{\sqrt [3]{54 x+\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 ) \sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+c_1){}^3}}}{6 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}\right ) (\log (x)+c_1)}{\sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+c_1){}^3}}} \\ y(x)\to -\frac {\sqrt [3]{2} \left (1-i \sqrt {3}\right ) (\log (x)+c_1)}{\sqrt [3]{54 x+\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*}