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 = 0.483575 (sec), leaf count = 65
\[\left \{\left \{y(x)\to \frac {1}{2} \left (c_1 x-\sqrt {-4 a+\left (4+c_1{}^2\right ) x^2}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\sqrt {-4 a+\left (4+c_1{}^2\right ) x^2}+c_1 x\right )\right \}\right \}\]
Maple ✓
cpu = 0.123 (sec), leaf count = 112
\[\left [\frac {1}{\frac {1}{y \left (x \right )^{2}}-\frac {1}{-x^{2}+a}} = -\frac {x \sqrt {x^{2}-a}}{\sqrt {\textit {\_C1} +\frac {4 a}{x^{2}-a}}}+\frac {x^{2}}{2}-\frac {a}{2}, \frac {1}{\frac {1}{y \left (x \right )^{2}}-\frac {1}{-x^{2}+a}} = \frac {x \sqrt {x^{2}-a}}{\sqrt {\textit {\_C1} +\frac {4 a}{x^{2}-a}}}+\frac {x^{2}}{2}-\frac {a}{2}\right ]\] 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))
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, 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]