Internal problem ID [3942]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 696.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {x \left (x^{3}-2 y^{3}\right ) y^{\prime }-\left (-y^{3}+2 x^{3}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 437
dsolve(x*(x^3-2*y(x)^3)*diff(y(x),x) = (2*x^3-y(x)^3)*y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {{\left (-12 x \left (9 c_{1} x^{2}-\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{6 c_{1}}+\frac {2 x}{{\left (-12 x \left (9 c_{1} x^{2}-\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}} y \left (x \right ) = -\frac {{\left (-12 x \left (9 c_{1} x^{2}-\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{12 c_{1}}-\frac {x}{{\left (-12 x \left (9 c_{1} x^{2}-\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {{\left (-12 x \left (9 c_{1} x^{2}-\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{6 c_{1}}-\frac {2 x}{{\left (-12 x \left (9 c_{1} x^{2}-\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}\right )}{2} y \left (x \right ) = -\frac {{\left (-12 x \left (9 c_{1} x^{2}-\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{12 c_{1}}-\frac {x}{{\left (-12 x \left (9 c_{1} x^{2}-\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {{\left (-12 x \left (9 c_{1} x^{2}-\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{6 c_{1}}-\frac {2 x}{{\left (-12 x \left (9 c_{1} x^{2}-\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} c_{1}^{3}-4\right )}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 60.353 (sec). Leaf size: 331
DSolve[x(x^3-2 y[x]^3)y'[x]==(2 x^3-y[x]^3)y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2} \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}+2 \sqrt [3]{3} e^{c_1} x}{6^{2/3} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} y(x)\to \frac {i \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}+3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} y(x)\to \frac {\sqrt [3]{2} \sqrt [6]{3} \left (-1-i \sqrt {3}\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}-3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \end{align*}