ODE
\[ x y'(x)=x \sqrt {x^2+y(x)^2}+y(x) \] ODE Classification
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Book solution method
Homogeneous equation, \(xy'(x)=x f(x) g(u)+y(x)\)
Mathematica ✓
cpu = 0.0184963 (sec), leaf count = 12
\[\left \{\left \{y(x)\to x \sinh \left (c_1+x\right )\right \}\right \}\]
Maple ✓
cpu = 2.555 (sec), leaf count = 28
\[ \left \{ \ln \left ( y \left ( x \right ) +\sqrt {{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) -x-\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x*y'[x] == y[x] + x*Sqrt[x^2 + y[x]^2],y[x],x]
Mathematica raw output
{{y[x] -> x*Sinh[x + C[1]]}}
Maple raw input
dsolve(x*diff(y(x),x) = y(x)+x*(x^2+y(x)^2)^(1/2), y(x),'implicit')
Maple raw output
ln(y(x)+(x^2+y(x)^2)^(1/2))-x-ln(x)-_C1 = 0