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]