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 = 0.0316742 (sec), leaf count = 17
\[\left \{\left \{y(x)\to x \cosh \left (c_1+i \log (x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.037 (sec), leaf count = 27
\[ \left \{ -\arctan \left ( {y \left ( x \right ) {\frac {1}{\sqrt {{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2}}}}} \right ) +\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] 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),'implicit')
Maple raw output
-arctan(1/(x^2-y(x)^2)^(1/2)*y(x))+ln(x)-_C1 = 0