4.13.24 $(2x^2+4xy(x)-y(x{)}^2)y^{\prime }(x)=x^2-4xy(x)-2y(x{)}^2$

ODE
$$(2x^2+4xy(x)-y(x{)}^2)y^{\prime }(x)=x^2-4xy(x)-2y(x{)}^2$$ ODE Classification

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

Book solution method
Exact equation

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

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

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

$$\{-\frac{1}{3}\ln \left(\frac{(x+y\left(x\right))(x^2-7xy\left(x\right)+{\left(y\left(x\right)\right)}^2)}{x^3}\right)-\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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