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 = 3.75411 (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 = 0.413 (sec), leaf count = 73

\[ \left \{ {\frac {1}{2} \left (2\,x\ln \left ({\frac {-2\,{x}^{2}+2\,\sqrt {-{x}^{2}}\sqrt { \left (y \relax (x ) \right ) ^{2}-{x}^{2}}}{y \relax (x ) }} \right ) + \left ({x}^{2}+ \left (y \relax (x ) \right ) ^{2}-2\,{\it \_C1} \right ) \sqrt {-{x}^{2}}-2\,x\ln \relax (x ) \right ) {\frac {1}{\sqrt {-{x}^{2}}}}}=0 \right \} \] 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),'implicit')

Maple raw output

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