4.13.24 \(\left (2 x^2+4 x y(x)-y(x)^2\right ) y'(x)=x^2-4 x y(x)-2 y(x)^2\)

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

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

Book solution method
Exact equation

Mathematica
cpu = 0.0341975 (sec), leaf count = 381

\[\left \{\left \{y(x)\to \frac {\sqrt [3]{\sqrt {54 e^{3 c_1} x^3+e^{6 c_1}-135 x^6}+e^{3 c_1}+27 x^3}}{\sqrt [3]{2}}+\frac {6 \sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {54 e^{3 c_1} x^3+e^{6 c_1}-135 x^6}+e^{3 c_1}+27 x^3}}+2 x\right \},\left \{y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {54 e^{3 c_1} x^3+e^{6 c_1}-135 x^6}+e^{3 c_1}+27 x^3}}{2 \sqrt [3]{2}}-\frac {3 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{\sqrt {54 e^{3 c_1} x^3+e^{6 c_1}-135 x^6}+e^{3 c_1}+27 x^3}}+2 x\right \},\left \{y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {54 e^{3 c_1} x^3+e^{6 c_1}-135 x^6}+e^{3 c_1}+27 x^3}}{2 \sqrt [3]{2}}+\frac {3 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{\sqrt {54 e^{3 c_1} x^3+e^{6 c_1}-135 x^6}+e^{3 c_1}+27 x^3}}+2 x\right \}\right \}\]

Maple
cpu = 0.016 (sec), leaf count = 35

\[ \left \{ -{\frac {1}{3}\ln \left ({\frac { \left (x+y \relax (x ) \right ) \left ({x}^{2}-7\,xy \relax (x ) + \left (y \relax (x ) \right ) ^{2} \right ) }{{x}^{3}}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[(2*x^2 + 4*x*y[x] - y[x]^2)*y'[x] == x^2 - 4*x*y[x] - 2*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> 2*x + (6*2^(1/3)*x^2)/(E^(3*C[1]) + 27*x^3 + Sqrt[E^(6*C[1]) + 54*E^(3
*C[1])*x^3 - 135*x^6])^(1/3) + (E^(3*C[1]) + 27*x^3 + Sqrt[E^(6*C[1]) + 54*E^(3*
C[1])*x^3 - 135*x^6])^(1/3)/2^(1/3)}, {y[x] -> 2*x - (3*2^(1/3)*(1 + I*Sqrt[3])*
x^2)/(E^(3*C[1]) + 27*x^3 + Sqrt[E^(6*C[1]) + 54*E^(3*C[1])*x^3 - 135*x^6])^(1/3
) + ((I/2)*(I + Sqrt[3])*(E^(3*C[1]) + 27*x^3 + Sqrt[E^(6*C[1]) + 54*E^(3*C[1])*
x^3 - 135*x^6])^(1/3))/2^(1/3)}, {y[x] -> 2*x + ((3*I)*2^(1/3)*(I + Sqrt[3])*x^2
)/(E^(3*C[1]) + 27*x^3 + Sqrt[E^(6*C[1]) + 54*E^(3*C[1])*x^3 - 135*x^6])^(1/3) -
 ((1 + I*Sqrt[3])*(E^(3*C[1]) + 27*x^3 + Sqrt[E^(6*C[1]) + 54*E^(3*C[1])*x^3 - 1
35*x^6])^(1/3))/(2*2^(1/3))}}

Maple raw input

dsolve((2*x^2+4*x*y(x)-y(x)^2)*diff(y(x),x) = x^2-4*x*y(x)-2*y(x)^2, y(x),'implicit')

Maple raw output

-1/3*ln((x+y(x))*(x^2-7*x*y(x)+y(x)^2)/x^3)-ln(x)-_C1 = 0