4.8.42 \(\sqrt {x^2+1} y'(x)=2 x-y(x)\)

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

[_linear]

Book solution method
Linear ODE

Mathematica
cpu = 0.0303187 (sec), leaf count = 33

\[\left \{\left \{y(x)\to e^{-\sinh ^{-1}(x)} \left (c_1+x^2+\sqrt {x^2+1} x-\sinh ^{-1}(x)\right )\right \}\right \}\]

Maple
cpu = 0.013 (sec), leaf count = 34

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

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

Mathematica raw output

{{y[x] -> (x^2 + x*Sqrt[1 + x^2] - ArcSinh[x] + C[1])/E^ArcSinh[x]}}

Maple raw input

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

Maple raw output

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