ODE
\[ y(x) \left (3 x^2+y(x)^2\right ) y'(x)+x \left (x^2+3 y(x)^2\right )=0 \] ODE Classification
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
Book solution method
Exact equation
Mathematica ✓
cpu = 0.0382626 (sec), leaf count = 129
\[\left \{\left \{y(x)\to -\sqrt {-\sqrt {e^{4 c_1}+8 x^4}-3 x^2}\right \},\left \{y(x)\to \sqrt {-\sqrt {e^{4 c_1}+8 x^4}-3 x^2}\right \},\left \{y(x)\to -\sqrt {\sqrt {e^{4 c_1}+8 x^4}-3 x^2}\right \},\left \{y(x)\to \sqrt {\sqrt {e^{4 c_1}+8 x^4}-3 x^2}\right \}\right \}\]
Maple ✓
cpu = 0.016 (sec), leaf count = 35
\[ \left \{ -{\frac {1}{4}\ln \left ( {\frac {{x}^{4}+6\,{x}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4}}{{x}^{4}}} \right ) }-\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x*(x^2 + 3*y[x]^2) + y[x]*(3*x^2 + y[x]^2)*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[-3*x^2 - Sqrt[E^(4*C[1]) + 8*x^4]]}, {y[x] -> Sqrt[-3*x^2 - Sqrt[E^(4*C[1]) + 8*x^4]]}, {y[x] -> -Sqrt[-3*x^2 + Sqrt[E^(4*C[1]) + 8*x^4]]}, {y[x] -> Sqrt[-3*x^2 + Sqrt[E^(4*C[1]) + 8*x^4]]}}
Maple raw input
dsolve((3*x^2+y(x)^2)*y(x)*diff(y(x),x)+x*(x^2+3*y(x)^2) = 0, y(x),'implicit')
Maple raw output
-1/4*ln((x^4+6*x^2*y(x)^2+y(x)^4)/x^4)-ln(x)-_C1 = 0