4.14.35 \(y(x) \left (a-3 x^2-y(x)^2\right ) y'(x)+x \left (a-x^2+y(x)^2\right )=0\)

ODE
\[ y(x) \left (a-3 x^2-y(x)^2\right ) y'(x)+x \left (a-x^2+y(x)^2\right )=0 \] ODE Classification

[_rational]

Book solution method
Change of Variable, Two new variables

Mathematica
cpu = 0.207636 (sec), leaf count = 39

\[\text {Solve}\left [\frac {1}{2} \left (\frac {a+2 y(x)^2}{x^2+y(x)^2}+\log \left (x^2+y(x)^2\right )\right )=c_1,y(x)\right ]\]

Maple
cpu = 0.191 (sec), leaf count = 47

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

DSolve[x*(a - x^2 + y[x]^2) + y[x]*(a - 3*x^2 - y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[(Log[x^2 + y[x]^2] + (a + 2*y[x]^2)/(x^2 + y[x]^2))/2 == C[1], y[x]]

Maple raw input

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

Maple raw output

1/(x^2+y(x)^2)*y(x)^2+a/(2*y(x)^2+2*x^2)+1/2*ln(x^2+y(x)^2)+_C1 = 0