2.556   ODE No. 556

1. Problem in Latex
2. Mathematica input
3. Maple input

$$x{y}^{\prime }(x{)}^2+\sqrt{y^{\prime }(x{)}^2+1}+y(x)=0$$ ✓ Mathematica : cpu = 5.32891 (sec), leaf count = 67

$$\text{Solve}[\{x=\frac{-\sqrt{K[1{]}^2+1}-{\sinh }^{-1}(K[1])}{(K[1]+1{)}^2}+\frac{c_1}{(K[1]+1{)}^2},y(x)=-xK[1{]}^2-\sqrt{K[1{]}^2+1}\},\{y(x),K[1]\}]$$ ✓ Maple : cpu = 0.213 (sec), leaf count = 581

$$\{\frac{c_1{x}^2}{{(-2x+\sqrt{-4xy\left(x\right)+2+2\sqrt{4x^2-4xy\left(x\right)+1}})}^2}+\frac{2(-2\mathrm{arcsinh}\left(\frac{\sqrt{-4xy\left(x\right)+2+2\sqrt{4x^2-4xy\left(x\right)+1}}}{2x}\right)+\sqrt{2}\sqrt{\frac{2x^2-2xy\left(x\right)+\sqrt{4x^2-4xy\left(x\right)+1}+1}{x^2}})x^2}{{(-2x+\sqrt{-4xy\left(x\right)+2+2\sqrt{4x^2-4xy\left(x\right)+1}})}^2}+x=0,\frac{c_1{x}^2}{{(2x+\sqrt{-4xy\left(x\right)+2+2\sqrt{4x^2-4xy\left(x\right)+1}})}^2}+\frac{2(2\mathrm{arcsinh}\left(\frac{\sqrt{-4xy\left(x\right)+2+2\sqrt{4x^2-4xy\left(x\right)+1}}}{2x}\right)+\sqrt{2}\sqrt{\frac{2x^2-2xy\left(x\right)+\sqrt{4x^2-4xy\left(x\right)+1}+1}{x^2}})x^2}{{(2x+\sqrt{-4xy\left(x\right)+2+2\sqrt{4x^2-4xy\left(x\right)+1}})}^2}+x=0,\frac{c_1{x}^2}{{(-2x+\sqrt{-4xy\left(x\right)-2\sqrt{4x^2-4xy\left(x\right)+1}+2})}^2}+\frac{2(-2\mathrm{arcsinh}\left(\frac{\sqrt{-4xy\left(x\right)-2\sqrt{4x^2-4xy\left(x\right)+1}+2}}{2x}\right)+\sqrt{\frac{4x^2-4xy\left(x\right)-2\sqrt{4x^2-4xy\left(x\right)+1}+2}{x^2}})x^2}{{(-2x+\sqrt{-4xy\left(x\right)-2\sqrt{4x^2-4xy\left(x\right)+1}+2})}^2}+x=0,\frac{c_1{x}^2}{{(2x+\sqrt{-4xy\left(x\right)-2\sqrt{4x^2-4xy\left(x\right)+1}+2})}^2}+\frac{2(2\mathrm{arcsinh}\left(\frac{\sqrt{-4xy\left(x\right)-2\sqrt{4x^2-4xy\left(x\right)+1}+2}}{2x}\right)+\sqrt{\frac{4x^2-4xy\left(x\right)-2\sqrt{4x^2-4xy\left(x\right)+1}+2}{x^2}})x^2}{{(2x+\sqrt{-4xy\left(x\right)-2\sqrt{4x^2-4xy\left(x\right)+1}+2})}^2}+x=0\}$$