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 = 0.0884609 (sec), leaf count = 48
\[\text {Solve}\left [\frac {2 a c_1-\log \left (-a+x^2+y(x)^2\right )+2 a \tan ^{-1}\left (\frac {y(x)}{x}\right )+\log \left (x^2+y(x)^2\right )}{a}=0,y(x)\right ]\]
Maple ✓
cpu = 0.151 (sec), leaf count = 40
\[ \left \{ -2\,\arctan \left ( {\frac {y \left ( x \right ) }{x}} \right ) a+\ln \left ( {x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}-a \right ) -\ln \left ( {x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2} \right ) -{\it \_C1}=0 \right \} \] 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),'implicit')
Maple raw output
-2*arctan(y(x)/x)*a+ln(x^2+y(x)^2-a)-ln(x^2+y(x)^2)-_C1 = 0