Internal problem ID [3395]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 23
Problem number: 656.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {3 x \left (x +y^{2}\right ) y^{\prime }+x^{3}-3 y x -2 y^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 450
dsolve(3*x*(x+y(x)^2)*diff(y(x),x)+x^3-3*x*y(x)-2*y(x)^3 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{2}-\frac {2 x}{\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (-4 c_{1} x^{2}-4 x^{3}+4 \sqrt {c_{1}^{2} x^{4}+2 c_{1} x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.15 (sec). Leaf size: 286
DSolve[3 x(x+y[x]^2)y'[x]+x^3-3 x y[x]-2 y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2 \sqrt [3]{2} x+2^{2/3} \left (\sqrt {x^3 \left (4+x (x-c_1){}^2\right )}+x^2 (-x+c_1)\right ){}^{2/3}}{2 \sqrt [3]{\sqrt {x^3 \left (4+x (x-c_1){}^2\right )}+x^2 (-x+c_1)}} \\ y(x)\to \frac {2 \sqrt [3]{-2} x+(-2)^{2/3} \left (\sqrt {x^3 \left (4+x (x-c_1){}^2\right )}+x^2 (-x+c_1)\right ){}^{2/3}}{2 \sqrt [3]{\sqrt {x^3 \left (4+x (x-c_1){}^2\right )}+x^2 (-x+c_1)}} \\ y(x)\to \frac {-\sqrt [3]{-2} \left (\sqrt {x^3 \left (4+x (x-c_1){}^2\right )}+x^2 (-x+c_1)\right ){}^{2/3}-i \sqrt {3} x+x}{2^{2/3} \sqrt [3]{\sqrt {x^3 \left (4+x (x-c_1){}^2\right )}+x^2 (-x+c_1)}} \\ \end{align*}