4.15.34 \(x \left (1-\sqrt {x^2-y(x)^2}\right ) y'(x)=y(x)\)

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

\[ \left \{ y \relax (x ) -\arctan \left ({y \relax (x ) {\frac {1}{\sqrt {{x}^{2}- \left (y \relax (x ) \right ) ^{2}}}}} \right ) -{\it \_C1}=0 \right \} \] 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