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 = 0.382495 (sec), leaf count = 67
\[\left \{\left \{y(x)\to \frac {e^{c_1}-\sqrt {-24 x^6+e^{2 c_1}}}{12 x^2}\right \},\left \{y(x)\to \frac {\sqrt {-24 x^6+e^{2 c_1}}+e^{c_1}}{12 x^2}\right \}\right \}\]
Maple ✓
cpu = 0.254 (sec), leaf count = 47
\[\left [y \left (x \right ) = -\frac {\textit {\_C1} \left (-1+\sqrt {-\frac {24 x^{6}}{\textit {\_C1}^{2}}+1}\right )}{12 x^{2}}, y \left (x \right ) = \frac {\textit {\_C1} \left (1+\sqrt {-\frac {24 x^{6}}{\textit {\_C1}^{2}}+1}\right )}{12 x^{2}}\right ]\] 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] + Sqrt[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))
Maple raw output
[y(x) = -1/12*_C1/x^2*(-1+(-24/_C1^2*x^6+1)^(1/2)), y(x) = 1/12*_C1/x^2*(1+(-24/_C1^2*x^6+1)^(1/2))]