##### 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 = 0.484653 (sec), leaf count = 381

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

Maple
cpu = 0.081 (sec), leaf count = 441

$\left [y \left (x \right ) = \frac {\frac {\left (108 x^{3} \textit {\_C1}^{3}+4+4 \sqrt {-135 \textit {\_C1}^{6} x^{6}+54 x^{3} \textit {\_C1}^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {12 \textit {\_C1}^{2} x^{2}}{\left (108 x^{3} \textit {\_C1}^{3}+4+4 \sqrt {-135 \textit {\_C1}^{6} x^{6}+54 x^{3} \textit {\_C1}^{3}+1}\right )^{\frac {1}{3}}}+2 x \textit {\_C1}}{\textit {\_C1}}, y \left (x \right ) = \frac {-\frac {\left (108 x^{3} \textit {\_C1}^{3}+4+4 \sqrt {-135 \textit {\_C1}^{6} x^{6}+54 x^{3} \textit {\_C1}^{3}+1}\right )^{\frac {1}{3}}}{4}-\frac {6 \textit {\_C1}^{2} x^{2}}{\left (108 x^{3} \textit {\_C1}^{3}+4+4 \sqrt {-135 \textit {\_C1}^{6} x^{6}+54 x^{3} \textit {\_C1}^{3}+1}\right )^{\frac {1}{3}}}+2 x \textit {\_C1} -\frac {i \sqrt {3}\, \left (\frac {\left (108 x^{3} \textit {\_C1}^{3}+4+4 \sqrt {-135 \textit {\_C1}^{6} x^{6}+54 x^{3} \textit {\_C1}^{3}+1}\right )^{\frac {1}{3}}}{2}-\frac {12 \textit {\_C1}^{2} x^{2}}{\left (108 x^{3} \textit {\_C1}^{3}+4+4 \sqrt {-135 \textit {\_C1}^{6} x^{6}+54 x^{3} \textit {\_C1}^{3}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\textit {\_C1}}, y \left (x \right ) = \frac {-\frac {\left (108 x^{3} \textit {\_C1}^{3}+4+4 \sqrt {-135 \textit {\_C1}^{6} x^{6}+54 x^{3} \textit {\_C1}^{3}+1}\right )^{\frac {1}{3}}}{4}-\frac {6 \textit {\_C1}^{2} x^{2}}{\left (108 x^{3} \textit {\_C1}^{3}+4+4 \sqrt {-135 \textit {\_C1}^{6} x^{6}+54 x^{3} \textit {\_C1}^{3}+1}\right )^{\frac {1}{3}}}+2 x \textit {\_C1} +\frac {i \sqrt {3}\, \left (\frac {\left (108 x^{3} \textit {\_C1}^{3}+4+4 \sqrt {-135 \textit {\_C1}^{6} x^{6}+54 x^{3} \textit {\_C1}^{3}+1}\right )^{\frac {1}{3}}}{2}-\frac {12 \textit {\_C1}^{2} x^{2}}{\left (108 x^{3} \textit {\_C1}^{3}+4+4 \sqrt {-135 \textit {\_C1}^{6} x^{6}+54 x^{3} \textit {\_C1}^{3}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\textit {\_C1}}\right ]$ 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))

Maple raw output

[y(x) = (1/2*(108*x^3*_C1^3+4+4*(-135*_C1^6*x^6+54*_C1^3*x^3+1)^(1/2))^(1/3)+12*
_C1^2*x^2/(108*x^3*_C1^3+4+4*(-135*_C1^6*x^6+54*_C1^3*x^3+1)^(1/2))^(1/3)+2*x*_C
1)/_C1, y(x) = (-1/4*(108*x^3*_C1^3+4+4*(-135*_C1^6*x^6+54*_C1^3*x^3+1)^(1/2))^(
1/3)-6*_C1^2*x^2/(108*x^3*_C1^3+4+4*(-135*_C1^6*x^6+54*_C1^3*x^3+1)^(1/2))^(1/3)
+2*x*_C1-1/2*I*3^(1/2)*(1/2*(108*x^3*_C1^3+4+4*(-135*_C1^6*x^6+54*_C1^3*x^3+1)^(
1/2))^(1/3)-12*_C1^2*x^2/(108*x^3*_C1^3+4+4*(-135*_C1^6*x^6+54*_C1^3*x^3+1)^(1/2
))^(1/3)))/_C1, y(x) = (-1/4*(108*x^3*_C1^3+4+4*(-135*_C1^6*x^6+54*_C1^3*x^3+1)^
(1/2))^(1/3)-6*_C1^2*x^2/(108*x^3*_C1^3+4+4*(-135*_C1^6*x^6+54*_C1^3*x^3+1)^(1/2
))^(1/3)+2*x*_C1+1/2*I*3^(1/2)*(1/2*(108*x^3*_C1^3+4+4*(-135*_C1^6*x^6+54*_C1^3*
x^3+1)^(1/2))^(1/3)-12*_C1^2*x^2/(108*x^3*_C1^3+4+4*(-135*_C1^6*x^6+54*_C1^3*x^3
+1)^(1/2))^(1/3)))/_C1]