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 = 1.29377 (sec), leaf count = 91
\[\text {Solve}\left [\frac {\sqrt {\frac {y(x)-x}{y(x)+x}} (y(x)+x) \left (\sqrt {1-\frac {y(x)^2}{x^2}}+2 \sin ^{-1}\left (\frac {\sqrt {1-\frac {y(x)}{x}}}{\sqrt {2}}\right )\right )}{x \sqrt {1-\frac {y(x)^2}{x^2}}}=c_1-i \log (x),y(x)\right ]\]
Maple ✓
cpu = 0.043 (sec), leaf count = 42
\[\left [\arctan \left (\frac {y \left (x \right )}{\sqrt {x^{2}-y \left (x \right )^{2}}}\right )-\frac {\sqrt {x^{2}-y \left (x \right )^{2}}}{x}+\ln \left (x \right )-\textit {\_C1} = 0\right ]\] 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[(Sqrt[(-x + y[x])/(x + y[x])]*(x + y[x])*(2*ArcSin[Sqrt[1 - y[x]/x]/Sqrt[2]] + Sqrt[1 - y[x]^2/x^2]))/(x*Sqrt[1 - y[x]^2/x^2]) == C[1] - I*Log[x], 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))
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]