Internal problem ID [3432]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 694.
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 (x^{3}-2 y^{3}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 355
dsolve(x*(2*x^3-y(x)^3)*diff(y(x),x) = (x^3-2*y(x)^3)*y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \left (\frac {\left (-108+8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2} c_{1}^{2}}{3 \left (-108+8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}+\frac {c_{1} x}{3}\right ) x \\ y \relax (x ) = \left (-\frac {\left (-108+8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{12}-\frac {x^{2} c_{1}^{2}}{3 \left (-108+8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}+\frac {c_{1} x}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (-108+8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2} c_{1}^{2}}{3 \left (-108+8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ y \relax (x ) = \left (-\frac {\left (-108+8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{12}-\frac {x^{2} c_{1}^{2}}{3 \left (-108+8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}+\frac {c_{1} x}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (-108+8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2} c_{1}^{2}}{3 \left (-108+8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 15.691 (sec). Leaf size: 474
DSolve[x(2 x^3-y[x]^3)y'[x]==(x^3-2 y[x]^3)y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{3} \left (e^{c_1} x^2+\sqrt [3]{e^{3 c_1} x^6-\frac {27 x^3}{2}+\frac {3}{2} \sqrt {81 x^6-12 e^{3 c_1} x^9}}+\frac {e^{2 c_1} x^4}{\sqrt [3]{e^{3 c_1} x^6-\frac {27 x^3}{2}+\frac {3}{2} \sqrt {81 x^6-12 e^{3 c_1} x^9}}}\right ) \\ y(x)\to \frac {1}{6} \left (2 e^{c_1} x^2+(-2)^{2/3} \sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}-\frac {2 \sqrt [3]{-2} e^{2 c_1} x^4}{\sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}\right ) \\ y(x)\to \frac {1}{3} e^{c_1} x^2 \left (1+\frac {e^{c_1} x^2 \text {Root}\left [\text {$\#$1}^3-2\&,3\right ]}{\sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}\right )-\frac {1}{3} \sqrt [3]{-\frac {1}{2}} \sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}} \\ y(x)\to \frac {\sqrt [3]{\sqrt {x^6}-x^3}}{\sqrt [3]{2}} \\ y(x)\to -\sqrt [3]{-\frac {1}{2}} \sqrt [3]{\sqrt {x^6}-x^3} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{\sqrt {x^6}-x^3}}{\sqrt [3]{2}} \\ \end{align*}