ODE
\[ \left (x \left (a-x^2-y(x)^2\right )+y(x)\right ) y'(x)-y(x) \left (a-x^2-y(x)^2\right )+x=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _rational]
Book solution method
Change of Variable, Two new variables
Mathematica ✓
cpu = 0.429233 (sec), leaf count = 48
\[\text {Solve}\left [\frac {-\log \left (-a+x^2+y(x)^2\right )+2 a \tan ^{-1}\left (\frac {y(x)}{x}\right )+2 a c_1+\log \left (x^2+y(x)^2\right )}{a}=0,y(x)\right ]\]
Maple ✓
cpu = 0.393 (sec), leaf count = 34
\[\left [y \left (x \right ) = \tan \left (\RootOf \left (2 a \textit {\_Z} +\ln \left (-\frac {x^{2}}{a \left (\cos ^{2}\left (\textit {\_Z} \right )\right )-x^{2}}\right )+\textit {\_C1} \right )\right ) x\right ]\] Mathematica raw input
DSolve[x - y[x]*(a - x^2 - y[x]^2) + (y[x] + x*(a - x^2 - y[x]^2))*y'[x] == 0,y[x],x]
Mathematica raw output
Solve[(2*a*ArcTan[y[x]/x] + 2*a*C[1] + Log[x^2 + y[x]^2] - Log[-a + x^2 + y[x]^2])/a == 0, y[x]]
Maple raw input
dsolve((x*(a-x^2-y(x)^2)+y(x))*diff(y(x),x)+x-(a-x^2-y(x)^2)*y(x) = 0, y(x))
Maple raw output
[y(x) = tan(RootOf(2*a*_Z+ln(-x^2/(a*cos(_Z)^2-x^2))+_C1))*x]