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

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.0692613 (sec), leaf count = 59

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )+\sinh ^{-1}(\text {$\#$1})\right )\& \right ]\left [c_1+\frac {1}{2} \left (-x \left (\sqrt {x^2+1}+x\right )-\sinh ^{-1}(x)\right )\right ]\right \}\right \}\]

Maple
cpu = 0.017 (sec), leaf count = 39

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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