4.12.40 \(3 x^4+8 x^3 y(x) y'(x)-6 x^2 y(x)^2-y(x)^4=0\)

ODE
\[ 3 x^4+8 x^3 y(x) y'(x)-6 x^2 y(x)^2-y(x)^4=0 \] ODE Classification

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

Book solution method
Homogeneous equation

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

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

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

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