ODE
\[ x \left (1-\sqrt {x^2-y(x)^2}\right ) y'(x)=y(x) \] ODE Classification
[`y=_G(x,y')`]
Book solution method
Exact equation, integrating factor
Mathematica ✓
cpu = 1.13682 (sec), leaf count = 40
\[\text {Solve}\left [c_1+i \log \left (\frac {2 \left (\sqrt {x^2-y(x)^2}-i y(x)\right )}{x}\right )=y(x),y(x)\right ]\]
Maple ✓
cpu = 0.283 (sec), leaf count = 27
\[ \left \{ y \left ( x \right ) -\arctan \left ( {y \left ( x \right ) {\frac {1}{\sqrt {{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}}}} \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x*(1 - Sqrt[x^2 - y[x]^2])*y'[x] == y[x],y[x],x]
Mathematica raw output
Solve[C[1] + I*Log[(2*((-I)*y[x] + Sqrt[x^2 - y[x]^2]))/x] == y[x], y[x]]
Maple raw input
dsolve(x*(1-(x^2-y(x)^2)^(1/2))*diff(y(x),x) = y(x), y(x),'implicit')
Maple raw output
y(x)-arctan(1/(x^2-y(x)^2)^(1/2)*y(x))-_C1 = 0