Internal problem ID [3938]
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]
\[ \boxed {x \left (x -y^{3}\right ) y^{\prime }-\left (3 x +y^{3}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.046 (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 \left (x \right ) = \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 \left (x \right ) = \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 \left (x \right ) = \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: 60.326 (sec). Leaf size: 356
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 {\frac {i \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}\right ){}^{2/3}}{x^2}-\left (\sqrt {3}+3 i\right ) e^{\frac {8 c_1}{3}}}{2\ 3^{5/6} \sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}} y(x)\to \frac {\frac {\left (-1-i \sqrt {3}\right ) \left (-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}\right ){}^{2/3}}{x^2}+i \sqrt [3]{3} \left (\sqrt {3}+i\right ) e^{\frac {8 c_1}{3}}}{2\ 3^{2/3} \sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}} \end{align*}