Internal problem ID [3369]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 22
Problem number: 629.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _exact, _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (3 x^{2}+2 y x +4 y^{2}\right ) y^{\prime }+2 x^{2}+6 y x +y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 431
dsolve((3*x^2+2*x*y(x)+4*y(x)^2)*diff(y(x),x)+2*x^2+6*x*y(x)+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\frac {\left (x^{3} c_{1}^{3}+8+2 \sqrt {333 x^{6} c_{1}^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}}{4}-\frac {11 x^{2} c_{1}^{2}}{4 \left (x^{3} c_{1}^{3}+8+2 \sqrt {333 x^{6} c_{1}^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}}-\frac {c_{1} x}{4}}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (x^{3} c_{1}^{3}+8+2 \sqrt {333 x^{6} c_{1}^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}}{8}+\frac {11 x^{2} c_{1}^{2}}{8 \left (x^{3} c_{1}^{3}+8+2 \sqrt {333 x^{6} c_{1}^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}}-\frac {c_{1} x}{4}-\frac {i \sqrt {3}\, \left (\frac {\left (x^{3} c_{1}^{3}+8+2 \sqrt {333 x^{6} c_{1}^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}}{4}+\frac {11 x^{2} c_{1}^{2}}{4 \left (x^{3} c_{1}^{3}+8+2 \sqrt {333 x^{6} c_{1}^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (x^{3} c_{1}^{3}+8+2 \sqrt {333 x^{6} c_{1}^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}}{8}+\frac {11 x^{2} c_{1}^{2}}{8 \left (x^{3} c_{1}^{3}+8+2 \sqrt {333 x^{6} c_{1}^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}}-\frac {c_{1} x}{4}+\frac {i \sqrt {3}\, \left (\frac {\left (x^{3} c_{1}^{3}+8+2 \sqrt {333 x^{6} c_{1}^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}}{4}+\frac {11 x^{2} c_{1}^{2}}{4 \left (x^{3} c_{1}^{3}+8+2 \sqrt {333 x^{6} c_{1}^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 49.757 (sec). Leaf size: 611
DSolve[(3 x^2+2 x y[x]+4 y[x]^2)y'[x]+2 x^2+6 x y[x]+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} \left (\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}-\frac {11 x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-x\right ) \\ y(x)\to \frac {1}{16} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}+\frac {22 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-4 x\right ) \\ y(x)\to \frac {1}{16} \left (\left (-2-2 i \sqrt {3}\right ) \sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}+\frac {22 \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-4 x\right ) \\ y(x)\to \frac {1}{4} \left (\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}-\frac {11 x^2}{\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}}-x\right ) \\ y(x)\to \frac {1}{8} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}+\frac {11 \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}}-2 x\right ) \\ y(x)\to \frac {1}{8} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}+\frac {11 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}}-2 x\right ) \\ \end{align*}