4.15.38 \(\left (x \sqrt {x^2+y(x)^2+1}-y(x) \left (x^2+y(x)^2\right )\right ) y'(x)=\sqrt {x^2+y(x)^2+1} y(x)+x \left (x^2+y(x)^2\right )\)

ODE
\[ \left (x \sqrt {x^2+y(x)^2+1}-y(x) \left (x^2+y(x)^2\right )\right ) y'(x)=\sqrt {x^2+y(x)^2+1} y(x)+x \left (x^2+y(x)^2\right ) \] ODE Classification

[[_1st_order, _with_linear_symmetries]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.126706 (sec), leaf count = 27

\[\text {Solve}\left [\sqrt {x^2+y(x)^2+1}+\tan ^{-1}\left (\frac {x}{y(x)}\right )=c_1,y(x)\right ]\]

Maple
cpu = 0.167 (sec), leaf count = 27

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

DSolve[(-(y[x]*(x^2 + y[x]^2)) + x*Sqrt[1 + x^2 + y[x]^2])*y'[x] == x*(x^2 + y[x]^2) + y[x]*Sqrt[1 + x^2 + y[x]^2],y[x],x]

Mathematica raw output

Solve[ArcTan[x/y[x]] + Sqrt[1 + x^2 + y[x]^2] == C[1], y[x]]

Maple raw input

dsolve((x*(1+x^2+y(x)^2)^(1/2)-y(x)*(x^2+y(x)^2))*diff(y(x),x) = x*(x^2+y(x)^2)+y(x)*(1+x^2+y(x)^2)^(1/2), y(x),'implicit')

Maple raw output

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