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

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

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