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

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

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