4.14.13 \(x \left (x^2-6 y(x)^2\right ) y'(x)=4 y(x) \left (x^2+3 y(x)^2\right )\)

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

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0366464 (sec), leaf count = 67

\[\left \{\left \{y(x)\to \frac {e^{c_1}-\sqrt {e^{2 c_1}-24 x^6}}{12 x^2}\right \},\left \{y(x)\to \frac {\sqrt {e^{2 c_1}-24 x^6}+e^{c_1}}{12 x^2}\right \}\right \}\]

Maple
cpu = 0.021 (sec), leaf count = 37

\[ \left \{ -{\frac {1}{3}\ln \left ({\frac {{x}^{2}+6\, \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) }+{\frac {1}{3}\ln \left ({\frac {y \relax (x ) }{x}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (E^C[1] - Sqrt[E^(2*C[1]) - 24*x^6])/(12*x^2)}, {y[x] -> (E^C[1] + Sqr
t[E^(2*C[1]) - 24*x^6])/(12*x^2)}}

Maple raw input

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

Maple raw output

-1/3*ln((x^2+6*y(x)^2)/x^2)+1/3*ln(y(x)/x)-ln(x)-_C1 = 0