Internal problem ID [3431]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 693.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {x \left (2 x^{3}+y^{3}\right ) y^{\prime }-\left (2 x^{3}-y x^{2}+y^{3}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 443
dsolve(x*(2*x^3+y(x)^3)*diff(y(x),x) = (2*x^3-x^2*y(x)+y(x)^3)*y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \left (\frac {\left (54+6 \sqrt {6 \ln \relax (x )^{3}+18 \ln \relax (x )^{2} c_{1}+18 \ln \relax (x ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{\frac {1}{3}}}{3}-\frac {3 \left (\frac {2 \ln \relax (x )}{3}+\frac {2 c_{1}}{3}\right )}{\left (54+6 \sqrt {6 \ln \relax (x )^{3}+18 \ln \relax (x )^{2} c_{1}+18 \ln \relax (x ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{\frac {1}{3}}}\right ) x \\ y \relax (x ) = \left (-\frac {\left (54+6 \sqrt {6 \ln \relax (x )^{3}+18 \ln \relax (x )^{2} c_{1}+18 \ln \relax (x ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}+\frac {\ln \relax (x )+c_{1}}{\left (54+6 \sqrt {6 \ln \relax (x )^{3}+18 \ln \relax (x )^{2} c_{1}+18 \ln \relax (x ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (54+6 \sqrt {6 \ln \relax (x )^{3}+18 \ln \relax (x )^{2} c_{1}+18 \ln \relax (x ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{\frac {1}{3}}}{3}+\frac {2 \ln \relax (x )+2 c_{1}}{\left (54+6 \sqrt {6 \ln \relax (x )^{3}+18 \ln \relax (x )^{2} c_{1}+18 \ln \relax (x ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ y \relax (x ) = \left (-\frac {\left (54+6 \sqrt {6 \ln \relax (x )^{3}+18 \ln \relax (x )^{2} c_{1}+18 \ln \relax (x ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}+\frac {\ln \relax (x )+c_{1}}{\left (54+6 \sqrt {6 \ln \relax (x )^{3}+18 \ln \relax (x )^{2} c_{1}+18 \ln \relax (x ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (54+6 \sqrt {6 \ln \relax (x )^{3}+18 \ln \relax (x )^{2} c_{1}+18 \ln \relax (x ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{\frac {1}{3}}}{3}+\frac {2 \ln \relax (x )+2 c_{1}}{\left (54+6 \sqrt {6 \ln \relax (x )^{3}+18 \ln \relax (x )^{2} c_{1}+18 \ln \relax (x ) c_{1}^{2}+6 c_{1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.893 (sec). Leaf size: 336
DSolve[x(2 x^3+y[x]^3)y'[x]==(2 x^3-x^2 y[x]+y[x]^3)y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {6^{2/3} x^2 (-\log (x)+c_1)+\sqrt [3]{6} \left (9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}\right ){}^{2/3}}{3 \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}{6^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) x^2 (\log (x)-c_1)}{\sqrt [3]{6} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}} \\ y(x)\to \frac {(-6)^{2/3} x^2 (-\log (x)+c_1)-\sqrt [3]{-6} \left (9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}\right ){}^{2/3}}{3 \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}} \\ \end{align*}