Internal problem ID [3430]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 692.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {x \left (x -y^{3}\right ) y^{\prime }-\left (3 x +y^{3}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.039 (sec). Leaf size: 348
dsolve(x*(x-y(x)^3)*diff(y(x),x) = (3*x+y(x)^3)*y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \left (\frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}}{3 x^{4}}+\frac {{\mathrm e}^{\frac {8 c_{1}}{3}}}{x^{4} \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}}\right ) x^{3} \\ y \relax (x ) = \left (-\frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}}{6 x^{4}}-\frac {{\mathrm e}^{\frac {8 c_{1}}{3}}}{2 x^{4} \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}}{3 x^{4}}-\frac {{\mathrm e}^{\frac {8 c_{1}}{3}}}{x^{4} \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x^{3} \\ y \relax (x ) = \left (-\frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}}{6 x^{4}}-\frac {{\mathrm e}^{\frac {8 c_{1}}{3}}}{2 x^{4} \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}}{3 x^{4}}-\frac {{\mathrm e}^{\frac {8 c_{1}}{3}}}{x^{4} \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x^{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.552 (sec). Leaf size: 435
DSolve[x(x-y[x]^3)y'[x]==(3 x+y[x]^3)y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{\frac {8 c_1}{3}}}{\sqrt [3]{-27 x^7+3 \sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}}+\frac {\sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}}{3^{2/3} x^2} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}}{3^{2/3} x^2}-\frac {\sqrt [3]{-\frac {1}{3}} e^{\frac {8 c_1}{3}}}{\sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}} \\ y(x)\to -\frac {\sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}} \left (\frac {e^{\frac {8 c_1}{3}} x^2 \text {Root}\left [\text {$\#$1}^3+24\&,2\right ]}{\left (-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}\right ){}^{2/3}}+i \sqrt {3}+1\right )}{2\ 3^{2/3} x^2} \\ y(x)\to \frac {\sqrt [3]{\sqrt {x^{14}}-x^7}}{x^2} \\ y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {x^{14}}-x^7}}{2 x^2} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {x^{14}}-x^7}}{2 x^2} \\ \end{align*}