ODE
\[ x y(x) \left (\sqrt {x^2-y(x)^2}+x\right ) y'(x)=x y(x)^2-\left (x^2-y(x)^2\right )^{3/2} \] ODE Classification
[[_1st_order, _with_linear_symmetries], _dAlembert]
Book solution method
Homogeneous equation
Mathematica ✗
cpu = 601.396 (sec), leaf count = 0 , timed out
$Aborted
Maple ✓
cpu = 0.065 (sec), leaf count = 35
\[\left [\frac {y \left (x \right )^{2}}{2 x^{2}}-\frac {\sqrt {x^{2}-y \left (x \right )^{2}}}{x}+\ln \left (x \right )-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[x*y[x]*(x + Sqrt[x^2 - y[x]^2])*y'[x] == x*y[x]^2 - (x^2 - y[x]^2)^(3/2),y[x],x]
Mathematica raw output
$Aborted
Maple raw input
dsolve(x*y(x)*(x+(x^2-y(x)^2)^(1/2))*diff(y(x),x) = x*y(x)^2-(x^2-y(x)^2)^(3/2), y(x))
Maple raw output
[1/2/x^2*y(x)^2-1/x*(x^2-y(x)^2)^(1/2)+ln(x)-_C1 = 0]