##### 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 = 0.745782 (sec), leaf count = 85

$\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {1}{2} \log \left (\sqrt {\text {\#1}^2+1}+\text {\#1}\right )-\frac {1}{4 \left (\sqrt {\text {\#1}^2+1}+\text {\#1}\right )^2}\& \right ]\left [-\frac {1}{4 \left (x-\sqrt {x^2+1}\right )^2}+\frac {1}{2} \log \left (x-\sqrt {x^2+1}\right )+c_1\right ]\right \}\right \}$

Maple
cpu = 0.018 (sec), leaf count = 39

$\left [\textit {\_C1} +x^{2}+x \sqrt {x^{2}+1}+\arcsinh \left (x \right )+y \left (x \right ) \sqrt {1+y \left (x \right )^{2}}+\arcsinh \left (y \left (x \right )\right )-y \left (x \right )^{2} = 0\right ]$ 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[Log[#1 + Sqrt[1 + #1^2]]/2 - 1/(4*(#1 + Sqrt[1 + #1^2]
)^2) & ][-1/4*1/(x - Sqrt[1 + x^2])^2 + C[1] + Log[x - Sqrt[1 + x^2]]/2]}}

Maple raw input

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

Maple raw output

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