4.13.32 \(\left (3 x^2+2 x y(x)+4 y(x)^2\right ) y'(x)+2 x^2+6 x y(x)+y(x)^2=0\)

ODE
\[ \left (3 x^2+2 x y(x)+4 y(x)^2\right ) y'(x)+2 x^2+6 x y(x)+y(x)^2=0 \] ODE Classification

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Book solution method
Exact equation

Mathematica
cpu = 0.0315862 (sec), leaf count = 382

\[\left \{\left \{y(x)\to \frac {1}{4} \left (\sqrt [3]{2 \sqrt {4 e^{3 c_1} x^3+16 e^{6 c_1}+333 x^6}+8 e^{3 c_1}+x^3}-\frac {11 x^2}{\sqrt [3]{2 \sqrt {4 e^{3 c_1} x^3+16 e^{6 c_1}+333 x^6}+8 e^{3 c_1}+x^3}}-x\right )\right \},\left \{y(x)\to \frac {1}{16} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{2 \sqrt {4 e^{3 c_1} x^3+16 e^{6 c_1}+333 x^6}+8 e^{3 c_1}+x^3}+\frac {22 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{2 \sqrt {4 e^{3 c_1} x^3+16 e^{6 c_1}+333 x^6}+8 e^{3 c_1}+x^3}}-4 x\right )\right \},\left \{y(x)\to \frac {1}{16} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{2 \sqrt {4 e^{3 c_1} x^3+16 e^{6 c_1}+333 x^6}+8 e^{3 c_1}+x^3}+\frac {22 \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{2 \sqrt {4 e^{3 c_1} x^3+16 e^{6 c_1}+333 x^6}+8 e^{3 c_1}+x^3}}-4 x\right )\right \}\right \}\]

Maple
cpu = 0.014 (sec), leaf count = 44

\[ \left \{ -{\frac {1}{3}\ln \left ({\frac {2\,{x}^{3}+9\,{x}^{2}y \relax (x ) +3\,x \left (y \relax (x ) \right ) ^{2}+4\, \left (y \relax (x ) \right ) ^{3}}{{x}^{3}}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

-1/3*ln((2*x^3+9*x^2*y(x)+3*x*y(x)^2+4*y(x)^3)/x^3)-ln(x)-_C1 = 0