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]
\[ \boxed {2 y^{2}+\left (4 y x +6 y^{2}\right ) y^{\prime }=-3 x^{2}} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 405
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 \left (x \right ) &= \frac {\frac {\left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 c_{1}^{6} x^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2} c_{1}^{2}}{\left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 c_{1}^{6} x^{6}-186 x^{3} c_{1}^{3}+81}\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 \sqrt {3}\, \left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 c_{1}^{6} x^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {2}{3}}-4 c_{1}^{2} x^{2}-4 c_{1} x \left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 c_{1}^{6} x^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}-\left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 c_{1}^{6} x^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {2}{3}}}{12 \left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 c_{1}^{6} x^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 c_{1}^{6} x^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {2}{3}}-4 x \left (i x c_{1} \sqrt {3}+c_{1} x +\left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 c_{1}^{6} x^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}}\right ) c_{1}}{12 \left (54-62 x^{3} c_{1}^{3}+6 \sqrt {105 c_{1}^{6} x^{6}-186 x^{3} c_{1}^{3}+81}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 39.668 (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*}