Internal problem ID [3806]
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`]]
\[ \boxed {x \left (2 x +3 y\right ) y^{\prime }-y^{2}=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 451
dsolve(x*(2*x+3*y(x))*diff(y(x),x) = y(x)^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\frac {\left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {3}\, \sqrt {-c_{1}^{2} x^{2} \left (4 c_{1}^{2} x^{2}-27\right )}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2} c_{1}^{2}}{\left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {3}\, \sqrt {-c_{1}^{2} x^{2} \left (4 c_{1}^{2} x^{2}-27\right )}\right )^{\frac {1}{3}}}-c_{1} x}{3 c_{1}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1}^{2} x^{2}-i \left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {3}\, \sqrt {-4 \left (c_{1}^{2} x^{2}-\frac {27}{4}\right ) c_{1}^{2} x^{2}}\right )^{\frac {2}{3}} \sqrt {3}-4 c_{1}^{2} x^{2}-4 \left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {3}\, \sqrt {-4 \left (c_{1}^{2} x^{2}-\frac {27}{4}\right ) c_{1}^{2} x^{2}}\right )^{\frac {1}{3}} c_{1} x -\left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {3}\, \sqrt {-4 \left (c_{1}^{2} x^{2}-\frac {27}{4}\right ) c_{1}^{2} x^{2}}\right )^{\frac {2}{3}}}{12 \left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {3}\, \sqrt {-4 \left (c_{1}^{2} x^{2}-\frac {27}{4}\right ) c_{1}^{2} x^{2}}\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {i \left (-4 c_{1}^{2} x^{2}+\left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {3}\, \sqrt {-4 \left (c_{1}^{2} x^{2}-\frac {27}{4}\right ) c_{1}^{2} x^{2}}\right )^{\frac {2}{3}}\right ) \sqrt {3}-{\left (2 c_{1} x +\left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {3}\, \sqrt {-4 \left (c_{1}^{2} x^{2}-\frac {27}{4}\right ) c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}\right )}^{2}}{12 \left (108 c_{1} x -8 x^{3} c_{1}^{3}+12 \sqrt {3}\, \sqrt {-4 \left (c_{1}^{2} x^{2}-\frac {27}{4}\right ) c_{1}^{2} x^{2}}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.159 (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*}