4.4.48 \(x y'(x)=\sqrt {x^2-y(x)^2}+y(x)\)

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

\[ \left \{ -\arctan \left ({y \relax (x ) {\frac {1}{\sqrt {{x}^{2}- \left (y \relax (x ) \right ) ^{2}}}}} \right ) +\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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