4.12.44 \(\sqrt {x^2+1} y(x) y'(x)+x \sqrt {y(x)^2+1}=0\)

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 = 0.019607 (sec), leaf count = 61

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

Maple
cpu = 0.019 (sec), leaf count = 20

\[ \left \{ \sqrt {{x}^{2}+1}+\sqrt {1+ \left (y \relax (x ) \right ) ^{2}}+{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

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