Internal problem ID [3298]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 20
Problem number: 554.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {x \left (2 x +3 y\right ) y^{\prime }-y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 461
dsolve(x*(2*x+3*y(x))*diff(y(x),x) = y(x)^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\frac {\left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{4} c_{1}^{4}+81 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2} c_{1}^{2}}{3 \left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{4} c_{1}^{4}+81 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {c_{1} x}{3}}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{4} c_{1}^{4}+81 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2} c_{1}^{2}}{3 \left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{4} c_{1}^{4}+81 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {c_{1} x}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{4} c_{1}^{4}+81 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2} c_{1}^{2}}{3 \left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{4} c_{1}^{4}+81 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{4} c_{1}^{4}+81 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2} c_{1}^{2}}{3 \left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{4} c_{1}^{4}+81 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {c_{1} x}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{4} c_{1}^{4}+81 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2} c_{1}^{2}}{3 \left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {-12 x^{4} c_{1}^{4}+81 x^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.132 (sec). Leaf size: 413
DSolve[x(2 x+3 y[x])y'[x]==y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{3} \left (\frac {x^2}{\sqrt [3]{-x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+\frac {27 e^{c_1} x}{2}}}+\sqrt [3]{-x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+\frac {27 e^{c_1} x}{2}}-x\right ) \\ y(x)\to \frac {1}{12} \left (-\frac {2 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{-x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+\frac {27 e^{c_1} x}{2}}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{-2 x^3+3 \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+27 e^{c_1} x}-4 x\right ) \\ y(x)\to \frac {1}{12} \left (\frac {2 i \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{-x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+\frac {27 e^{c_1} x}{2}}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-2 x^3+3 \sqrt {3} \sqrt {e^{c_1} x^2 \left (-4 x^2+27 e^{c_1}\right )}+27 e^{c_1} x}-4 x\right ) \\ \end{align*}