60.2.153 problem 729

Internal problem ID [10740]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 729
Date solved : Monday, January 27, 2025 at 09:36:55 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: 316

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 \ln \left (x \right )^{3}+72 c_{1} \ln \left (x \right )^{2}-72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}+6 \ln \left (x \right )-6 c_{1}}{3 \left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 c_{1} \ln \left (x \right )^{2}-72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{{1}/{3}}} \\ y &= -\frac {\left (\frac {i \sqrt {3}}{6}+\frac {1}{6}\right ) \left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 c_{1} \ln \left (x \right )^{2}-72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}+\left (i \sqrt {3}-1\right ) \left (-\ln \left (x \right )+c_{1} \right )}{\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 c_{1} \ln \left (x \right )^{2}-72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{{1}/{3}}} \\ y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 c_{1} \ln \left (x \right )^{2}-72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}}{6}+\left (1+i \sqrt {3}\right ) \left (-\ln \left (x \right )+c_{1} \right )}{\left (-27 x +3 \sqrt {-24 \ln \left (x \right )^{3}+72 c_{1} \ln \left (x \right )^{2}-72 \ln \left (x \right ) c_{1}^{2}+24 c_{1}^{3}+81 x^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 6.507 (sec). Leaf size: 320

DSolve[D[y[x],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*}