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

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 = 0.0195762 (sec), leaf count = 13

\[\left \{\left \{y(x)\to x \sinh \left (c_1+\log (x)\right )\right \}\right \}\]

Maple
cpu = 0.037 (sec), leaf count = 27

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

DSolve[x*y'[x] == y[x] + Sqrt[x^2 + y[x]^2],y[x],x]

Mathematica raw output

{{y[x] -> x*Sinh[C[1] + 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

1/x^2*y(x)+1/x^2*(x^2+y(x)^2)^(1/2)-_C1 = 0