ODE
\[ \left (3 x^2+2 x y(x)+4 y(x)^2\right ) y'(x)+2 x^2+6 x y(x)+y(x)^2=0 \] ODE Classification
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
Book solution method
Exact equation
Mathematica ✓
cpu = 0.403899 (sec), leaf count = 382
\[\left \{\left \{y(x)\to \frac {1}{4} \left (\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}-\frac {11 x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-x\right )\right \},\left \{y(x)\to \frac {1}{16} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}+\frac {22 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-4 x\right )\right \},\left \{y(x)\to \frac {1}{16} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}+\frac {22 \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-4 x\right )\right \}\right \}\]
Maple ✓
cpu = 0.219 (sec), leaf count = 431
\[\left [y \left (x \right ) = \frac {\frac {\left (x^{3} \textit {\_C1}^{3}+8+2 \sqrt {333 \textit {\_C1}^{6} x^{6}+4 x^{3} \textit {\_C1}^{3}+16}\right )^{\frac {1}{3}}}{4}-\frac {11 \textit {\_C1}^{2} x^{2}}{4 \left (x^{3} \textit {\_C1}^{3}+8+2 \sqrt {333 \textit {\_C1}^{6} x^{6}+4 x^{3} \textit {\_C1}^{3}+16}\right )^{\frac {1}{3}}}-\frac {x \textit {\_C1}}{4}}{\textit {\_C1}}, y \left (x \right ) = \frac {-\frac {\left (x^{3} \textit {\_C1}^{3}+8+2 \sqrt {333 \textit {\_C1}^{6} x^{6}+4 x^{3} \textit {\_C1}^{3}+16}\right )^{\frac {1}{3}}}{8}+\frac {11 \textit {\_C1}^{2} x^{2}}{8 \left (x^{3} \textit {\_C1}^{3}+8+2 \sqrt {333 \textit {\_C1}^{6} x^{6}+4 x^{3} \textit {\_C1}^{3}+16}\right )^{\frac {1}{3}}}-\frac {x \textit {\_C1}}{4}-\frac {i \sqrt {3}\, \left (\frac {\left (x^{3} \textit {\_C1}^{3}+8+2 \sqrt {333 \textit {\_C1}^{6} x^{6}+4 x^{3} \textit {\_C1}^{3}+16}\right )^{\frac {1}{3}}}{4}+\frac {11 \textit {\_C1}^{2} x^{2}}{4 \left (x^{3} \textit {\_C1}^{3}+8+2 \sqrt {333 \textit {\_C1}^{6} x^{6}+4 x^{3} \textit {\_C1}^{3}+16}\right )^{\frac {1}{3}}}\right )}{2}}{\textit {\_C1}}, y \left (x \right ) = \frac {-\frac {\left (x^{3} \textit {\_C1}^{3}+8+2 \sqrt {333 \textit {\_C1}^{6} x^{6}+4 x^{3} \textit {\_C1}^{3}+16}\right )^{\frac {1}{3}}}{8}+\frac {11 \textit {\_C1}^{2} x^{2}}{8 \left (x^{3} \textit {\_C1}^{3}+8+2 \sqrt {333 \textit {\_C1}^{6} x^{6}+4 x^{3} \textit {\_C1}^{3}+16}\right )^{\frac {1}{3}}}-\frac {x \textit {\_C1}}{4}+\frac {i \sqrt {3}\, \left (\frac {\left (x^{3} \textit {\_C1}^{3}+8+2 \sqrt {333 \textit {\_C1}^{6} x^{6}+4 x^{3} \textit {\_C1}^{3}+16}\right )^{\frac {1}{3}}}{4}+\frac {11 \textit {\_C1}^{2} x^{2}}{4 \left (x^{3} \textit {\_C1}^{3}+8+2 \sqrt {333 \textit {\_C1}^{6} x^{6}+4 x^{3} \textit {\_C1}^{3}+16}\right )^{\frac {1}{3}}}\right )}{2}}{\textit {\_C1}}\right ]\] Mathematica raw input
DSolve[2*x^2 + 6*x*y[x] + y[x]^2 + (3*x^2 + 2*x*y[x] + 4*y[x]^2)*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-x - (11*x^2)/(8*E^(3*C[1]) + x^3 + 2*Sqrt[16*E^(6*C[1]) + 4*E^(3*C[1])*x^3 + 333*x^6])^(1/3) + (8*E^(3*C[1]) + x^3 + 2*Sqrt[16*E^(6*C[1]) + 4*E^(3*C[1])*x^3 + 333*x^6])^(1/3))/4}, {y[x] -> (-4*x + (22*(1 + I*Sqrt[3])*x^2)/(8*E^(3*C[1]) + x^3 + 2*Sqrt[16*E^(6*C[1]) + 4*E^(3*C[1])*x^3 + 333*x^6])^(1/3) + (2*I)*(I + Sqrt[3])*(8*E^(3*C[1]) + x^3 + 2*Sqrt[16*E^(6*C[1]) + 4*E^(3*C[1])*x^3 + 333*x^6])^(1/3))/16}, {y[x] -> (-4*x + (22*(1 - I*Sqrt[3])*x^2)/(8*E^(3*C[1]) + x^3 + 2*Sqrt[16*E^(6*C[1]) + 4*E^(3*C[1])*x^3 + 333*x^6])^(1/3) - 2*(1 + I*Sqrt[3])*(8*E^(3*C[1]) + x^3 + 2*Sqrt[16*E^(6*C[1]) + 4*E^(3*C[1])*x^3 + 333*x^6])^(1/3))/16}}
Maple raw input
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))
Maple raw output
[y(x) = (1/4*(x^3*_C1^3+8+2*(333*_C1^6*x^6+4*_C1^3*x^3+16)^(1/2))^(1/3)-11/4*_C1^2*x^2/(x^3*_C1^3+8+2*(333*_C1^6*x^6+4*_C1^3*x^3+16)^(1/2))^(1/3)-1/4*x*_C1)/_C1, y(x) = (-1/8*(x^3*_C1^3+8+2*(333*_C1^6*x^6+4*_C1^3*x^3+16)^(1/2))^(1/3)+11/8*_C1^2*x^2/(x^3*_C1^3+8+2*(333*_C1^6*x^6+4*_C1^3*x^3+16)^(1/2))^(1/3)-1/4*x*_C1-1/2*I*3^(1/2)*(1/4*(x^3*_C1^3+8+2*(333*_C1^6*x^6+4*_C1^3*x^3+16)^(1/2))^(1/3)+11/4*_C1^2*x^2/(x^3*_C1^3+8+2*(333*_C1^6*x^6+4*_C1^3*x^3+16)^(1/2))^(1/3)))/_C1, y(x) = (-1/8*(x^3*_C1^3+8+2*(333*_C1^6*x^6+4*_C1^3*x^3+16)^(1/2))^(1/3)+11/8*_C1^2*x^2/(x^3*_C1^3+8+2*(333*_C1^6*x^6+4*_C1^3*x^3+16)^(1/2))^(1/3)-1/4*x*_C1+1/2*I*3^(1/2)*(1/4*(x^3*_C1^3+8+2*(333*_C1^6*x^6+4*_C1^3*x^3+16)^(1/2))^(1/3)+11/4*_C1^2*x^2/(x^3*_C1^3+8+2*(333*_C1^6*x^6+4*_C1^3*x^3+16)^(1/2))^(1/3)))/_C1]