ODE
\[ x y'(x)=\sqrt {x^2-y(x)^2}+y(x) \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
Homogeneous equation
Mathematica ✓
cpu = 0.262795 (sec), leaf count = 17
\[\{\{y(x)\to x \cosh (i \log (x)+c_1)\}\}\]
Maple ✓
cpu = 0.049 (sec), leaf count = 27
\[\left [-\arctan \left (\frac {y \left (x \right )}{\sqrt {x^{2}-y \left (x \right )^{2}}}\right )+\ln \left (x \right )-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[x*y'[x] == y[x] + Sqrt[x^2 - y[x]^2],y[x],x]
Mathematica raw output
{{y[x] -> x*Cosh[C[1] + I*Log[x]]}}
Maple raw input
dsolve(x*diff(y(x),x) = y(x)+(x^2-y(x)^2)^(1/2), y(x))
Maple raw output
[-arctan(1/(x^2-y(x)^2)^(1/2)*y(x))+ln(x)-_C1 = 0]