Internal problem ID [111]
Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 1.6, Substitution methods and exact equations. Page 74
Problem number: 33.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _exact, _rational, _dAlembert]
Solve \begin {gather*} \boxed {3 x^{2}+2 y^{2}+\left (4 y x +6 y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 441
dsolve(3*x^2+2*y(x)^2+(4*x*y(x)+6*y(x)^2)*diff(y(x),x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\frac {\left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 x^{6} c_{1}^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2} c_{1}^{2}}{3 \left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 x^{6} c_{1}^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}-\frac {x c_{1}}{3}}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 x^{6} c_{1}^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{12}-\frac {x^{2} c_{1}^{2}}{3 \left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 x^{6} c_{1}^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}-\frac {x c_{1}}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 x^{6} c_{1}^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2} c_{1}^{2}}{3 \left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 x^{6} c_{1}^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 x^{6} c_{1}^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{12}-\frac {x^{2} c_{1}^{2}}{3 \left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 x^{6} c_{1}^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}-\frac {x c_{1}}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 x^{6} c_{1}^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2} c_{1}^{2}}{3 \left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 x^{6} c_{1}^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 42.33 (sec). Leaf size: 679
DSolve[3*x^2+2*y[x]^2+(4*x*y[x]+6*y[x]^2)*y'[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-124 x^3+\sqrt {-256 x^6+\left (-124 x^3+108 e^{2 c_1}\right ){}^2}+108 e^{2 c_1}}}{6 \sqrt [3]{2}}+\frac {2 \sqrt [3]{2} x^2}{3 \sqrt [3]{-124 x^3+\sqrt {-256 x^6+\left (-124 x^3+108 e^{2 c_1}\right ){}^2}+108 e^{2 c_1}}}-\frac {x}{3} \\ y(x)\to \frac {1}{12} i \left (\sqrt {3}+i\right ) \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}-\frac {i \left (\sqrt {3}-i\right ) x^2}{3 \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}}-\frac {x}{3} \\ y(x)\to -\frac {1}{12} i \left (\sqrt {3}-i\right ) \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{3 \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}}-\frac {x}{3} \\ y(x)\to \frac {1}{6} \left (\sqrt [3]{6 \sqrt {105} \sqrt {x^6}-62 x^3}+\frac {2\ 2^{2/3} x^2}{\sqrt [3]{3 \sqrt {105} \sqrt {x^6}-31 x^3}}-2 x\right ) \\ y(x)\to \frac {1}{12} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{6 \sqrt {105} \sqrt {x^6}-62 x^3}-\frac {2 i 2^{2/3} \left (\sqrt {3}-i\right ) x^2}{\sqrt [3]{3 \sqrt {105} \sqrt {x^6}-31 x^3}}-4 x\right ) \\ y(x)\to \frac {1}{12} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{6 \sqrt {105} \sqrt {x^6}-62 x^3}+\frac {2 i 2^{2/3} \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{3 \sqrt {105} \sqrt {x^6}-31 x^3}}-4 x\right ) \\ \end{align*}