ODE
\[ \sqrt {x^2+1} y(x) y'(x)+x \sqrt {y(x)^2+1}=0 \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.400372 (sec), leaf count = 61
\[\left \{\left \{y(x)\to -\sqrt {x^2+c_1 \left (-2 \sqrt {x^2+1}+c_1\right )}\right \},\left \{y(x)\to \sqrt {x^2+c_1 \left (-2 \sqrt {x^2+1}+c_1\right )}\right \}\right \}\]
Maple ✓
cpu = 0.015 (sec), leaf count = 20
\[\left [\sqrt {x^{2}+1}+\sqrt {1+y \left (x \right )^{2}}+\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[x*Sqrt[1 + y[x]^2] + Sqrt[1 + x^2]*y[x]*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[x^2 + C[1]*(-2*Sqrt[1 + x^2] + C[1])]}, {y[x] -> Sqrt[x^2 + C[1]*(-2*Sqrt[1 + x^2] + C[1])]}}
Maple raw input
dsolve(y(x)*diff(y(x),x)*(x^2+1)^(1/2)+x*(1+y(x)^2)^(1/2) = 0, y(x))
Maple raw output
[(x^2+1)^(1/2)+(1+y(x)^2)^(1/2)+_C1 = 0]