##### 4.15.37 $$\left (x-y(x)^2 \sqrt {y(x)^2-x^2}\right ) y'(x)=y(x) \left (x \sqrt {y(x)^2-x^2}+1\right )$$

ODE
$\left (x-y(x)^2 \sqrt {y(x)^2-x^2}\right ) y'(x)=y(x) \left (x \sqrt {y(x)^2-x^2}+1\right )$ ODE Classiﬁcation

[NONE]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 3.1294 (sec), leaf count = 35

$\text {Solve}\left [2 \tan ^{-1}\left (\frac {x}{\sqrt {y(x)^2-x^2}}\right )+x^2+y(x)^2=2 c_1,y(x)\right ]$

Maple
cpu = 0.424 (sec), leaf count = 73

$\left [\frac {y \left (x \right )^{2}}{2}+\frac {x \ln \left (\frac {-2 x^{2}+2 \sqrt {-x^{2}}\, \sqrt {y \left (x \right )^{2}-x^{2}}}{y \left (x \right )}\right )}{\sqrt {-x^{2}}}+\frac {\sqrt {-x^{2}}\, \ln \left (x \right )}{x}+\frac {x^{2}}{2}-\textit {\_C1} = 0\right ]$ Mathematica raw input

DSolve[(x - y[x]^2*Sqrt[-x^2 + y[x]^2])*y'[x] == y[x]*(1 + x*Sqrt[-x^2 + y[x]^2]),y[x],x]

Mathematica raw output

Solve[x^2 + 2*ArcTan[x/Sqrt[-x^2 + y[x]^2]] + y[x]^2 == 2*C[1], y[x]]

Maple raw input

dsolve((x-y(x)^2*(y(x)^2-x^2)^(1/2))*diff(y(x),x) = (1+x*(y(x)^2-x^2)^(1/2))*y(x), y(x))

Maple raw output

[1/2*y(x)^2+x/(-x^2)^(1/2)*ln((-2*x^2+2*(-x^2)^(1/2)*(y(x)^2-x^2)^(1/2))/y(x))+(
-x^2)^(1/2)/x*ln(x)+1/2*x^2-_C1 = 0]