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

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

[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Book solution method
Homogeneous equation, special

Mathematica
cpu = 0.0290743 (sec), leaf count = 65

\[\left \{\left \{y(x)\to \frac {1}{2} \left (c_1 x-\sqrt {\left (c_1^2+4\right ) x^2-4 a}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\sqrt {\left (c_1^2+4\right ) x^2-4 a}+c_1 x\right )\right \}\right \}\]

Maple
cpu = 0.092 (sec), leaf count = 114

\[ \left \{ \left (\left (y \relax (x ) \right ) ^{-2}- \left (-{x}^{2}+a \right ) ^{-1} \right ) ^{-1}-{x\sqrt {{x}^{2}-a}{\frac {1}{\sqrt {{\it \_C1}+4\,{\frac {a}{{x}^{2}-a}}}}}}-{\frac {{x}^{2}}{2}}+{\frac {a}{2}}=0, \left (\left (y \relax (x ) \right ) ^{-2}- \left (-{x}^{2}+a \right ) ^{-1} \right ) ^{-1}+{x\sqrt {{x}^{2}-a}{\frac {1}{\sqrt {{\it \_C1}+4\,{\frac {a}{{x}^{2}-a}}}}}}-{\frac {{x}^{2}}{2}}+{\frac {a}{2}}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (x*C[1] - Sqrt[-4*a + x^2*(4 + C[1]^2)])/2}, {y[x] -> (x*C[1] + Sqrt[-
4*a + x^2*(4 + C[1]^2)])/2}}

Maple raw input

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

Maple raw output

1/(1/y(x)^2-1/(-x^2+a))+x*(x^2-a)^(1/2)/(_C1+4*a/(x^2-a))^(1/2)-1/2*x^2+1/2*a = 
0, 1/(1/y(x)^2-1/(-x^2+a))-x*(x^2-a)^(1/2)/(_C1+4*a/(x^2-a))^(1/2)-1/2*x^2+1/2*a
 = 0