ODE
\[ x \sqrt {x^2-y(x)^2}+x (y(x)+x) y'(x)-y(x) (y(x)+x)=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
Homogeneous equation
Mathematica ✓
cpu = 0.214498 (sec), leaf count = 74
\[\text {Solve}\left [\frac {y(x) \sqrt {\frac {y(x)-x}{y(x)+x}}}{x}+\sqrt {\frac {y(x)-x}{y(x)+x}}+2 \sinh ^{-1}\left (\frac {\sqrt {\frac {y(x)}{x}-1}}{\sqrt {2}}\right )+i \log (x)=c_1,y(x)\right ]\]
Maple ✓
cpu = 0.041 (sec), leaf count = 42
\[ \left \{ \arctan \left ( {y \left ( x \right ) {\frac {1}{\sqrt {{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}}}} \right ) -{\frac {1}{x}\sqrt {{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}}+\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[-(y[x]*(x + y[x])) + x*Sqrt[x^2 - y[x]^2] + x*(x + y[x])*y'[x] == 0,y[x],x]
Mathematica raw output
Solve[2*ArcSinh[Sqrt[-1 + y[x]/x]/Sqrt[2]] + I*Log[x] + Sqrt[(-x + y[x])/(x + y[x])] + (y[x]*Sqrt[(-x + y[x])/(x + y[x])])/x == C[1], y[x]]
Maple raw input
dsolve(x*(x+y(x))*diff(y(x),x)-(x+y(x))*y(x)+x*(x^2-y(x)^2)^(1/2) = 0, y(x),'implicit')
Maple raw output
arctan(1/(x^2-y(x)^2)^(1/2)*y(x))-1/x*(x^2-y(x)^2)^(1/2)+ln(x)-_C1 = 0