4.12.40 $3x^4+8x^{3}y(x)y^{\prime }(x)-6x^{2}y(x{)}^2-y(x{)}^4=0$

ODE
$$3x^4+8x^{3}y(x)y^{\prime }(x)-6x^{2}y(x{)}^2-y(x{)}^4=0$$ ODE Classification

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

Book solution method
Homogeneous equation

Mathematica ✓
cpu = 0.0286494 (sec), leaf count = 78

$$\{\{y(x)\to -\frac{\sqrt{-x^2(e^{8c_1}x+3)}}{\sqrt{e^{8c_1}x-1}}\},\{y(x)\to \frac{\sqrt{-x^2(e^{8c_1}x+3)}}{\sqrt{e^{8c_1}x-1}}\}\}$$

Maple ✓
cpu = 0.024 (sec), leaf count = 41

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

DSolve[3*x^4 - 6*x^2*y[x]^2 - y[x]^4 + 8*x^3*y[x]*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[-(x^2*(3 + E^(8*C[1])*x))]/Sqrt[-1 + E^(8*C[1])*x])}, {y[x] -> 
Sqrt[-(x^2*(3 + E^(8*C[1])*x))]/Sqrt[-1 + E^(8*C[1])*x]}}

Maple raw input

dsolve(8*x^3*y(x)*diff(y(x),x)+3*x^4-6*x^2*y(x)^2-y(x)^4 = 0, y(x),'implicit')

Maple raw output

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