Internal problem ID [3394]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 23
Problem number: 655.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _exact, _rational]
Solve \begin {gather*} \boxed {x \left (-3 y^{2}+x \right ) y^{\prime }+\left (2 x -y^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 327
dsolve(x*(x-3*y(x)^2)*diff(y(x),x)+(2*x-y(x)^2)*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {2 x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {2 x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {2 x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 30.296 (sec). Leaf size: 328
DSolve[x(x-3 y[x]^2)y'[x]+(2 x-y[x]^2)y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 \sqrt [3]{3} x^3+\sqrt [3]{2} \left (9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{6^{2/3} x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ y(x)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^3+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ y(x)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^3+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ \end{align*}