2.556   ODE No. 556

\[ x y'(x)^2+\sqrt {y'(x)^2+1}+y(x)=0 \] Mathematica : cpu = 4.18701 (sec), leaf count = 67


\[\text {Solve}\left [\left \{x=\frac {-\sqrt {K[1]^2+1}-\sinh ^{-1}(K[1])}{(K[1]+1)^2}+\frac {c_1}{(K[1]+1)^2},y(x)=-x K[1]^2-\sqrt {K[1]^2+1}\right \},\{y(x),K[1]\}\right ]\] Maple : cpu = 0.232 (sec), leaf count = 581


\[\frac {x^{2} c_{1}}{\left (\sqrt {-4 x y \relax (x )+2+2 \sqrt {4 x^{2}-4 x y \relax (x )+1}}-2 x \right )^{2}}+x +\frac {2 x^{2} \left (\sqrt {2}\, \sqrt {\frac {2 x^{2}-2 x y \relax (x )+\sqrt {4 x^{2}-4 x y \relax (x )+1}+1}{x^{2}}}-2 \arcsinh \left (\frac {\sqrt {-4 x y \relax (x )+2+2 \sqrt {4 x^{2}-4 x y \relax (x )+1}}}{2 x}\right )\right )}{\left (\sqrt {-4 x y \relax (x )+2+2 \sqrt {4 x^{2}-4 x y \relax (x )+1}}-2 x \right )^{2}} = 0\]