60.2.172 problem 748

Internal problem ID [10759]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 748
Date solved : Monday, January 27, 2025 at 09:41:26 PM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=\frac {y \left (x +y\right )}{x \left (x +y^{3}\right )} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 312

dsolve(diff(y(x),x) = y(x)*(x+y(x))/x/(x+y(x)^3),y(x), singsol=all)
 

\begin{align*} y &= \frac {\left (27 x +3 \sqrt {-24 c_{1}^{3}-72 \ln \left (x \right ) c_{1}^{2}-72 c_{1} \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{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 c_{1} \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{1}/{3}}} \\ y &= \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 c_{1} \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{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 c_{1} \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{1}/{3}}} \\ y &= \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 c_{1} \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{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 c_{1} \ln \left (x \right )^{2}-24 \ln \left (x \right )^{3}+81 x^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 9.632 (sec). Leaf size: 300

DSolve[D[y[x],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)+1+c_1)}{\sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+1+c_1){}^3}}}+\frac {\sqrt [3]{9 x+\frac {1}{6} \sqrt {2916 x^2-864 (\log (x)+1+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)+1+c_1){}^3}}}{6 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}\right ) (\log (x)+1+c_1)}{\sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+1+c_1){}^3}}} \\ y(x)\to -\frac {\sqrt [3]{2} \left (1-i \sqrt {3}\right ) (\log (x)+1+c_1)}{\sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+1+c_1){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{54 x+\sqrt {2916 x^2-864 (\log (x)+1+c_1){}^3}}}{6 \sqrt [3]{2}} \\ y(x)\to 0 \\ \end{align*}