2.516   ODE No. 516

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

$$(x^2+y(x{)}^2)f\left(\frac{x}{\sqrt{x^2+y(x{)}^2}}\right)(y^{\prime }(x{)}^2+1)-{(x{y}^{\prime }(x)-y(x))}^2=0$$ ✓ Mathematica : cpu = 0.442299 (sec), leaf count = 253

$$\{\text{Solve}[{\int }_1^{\frac{y(x)}{x}}\frac{f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)K[1{]}^2+f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)-1}{\sqrt{f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)}(K[1]-i)(K[1]+i)(\sqrt{f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)}K[1]+i\sqrt{f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)-1})}dK[1]=-\log (x)+c_1,y(x)],\text{Solve}[{\int }_1^{\frac{y(x)}{x}}\frac{f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)K[2{]}^2+f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)-1}{\sqrt{f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)}(K[2]-i)(K[2]+i)(\sqrt{f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)}K[2]-i\sqrt{f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)-1})}dK[2]=-\log (x)+c_1,y(x)]\}$$ ✓ Maple : cpu = 2.997 (sec), leaf count = 72

$$\{y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-\ln \left(x\right)+{\int }^{\mathit{\_}\mathit{Z}}-\frac{1}{{\mathit{\_}\mathit{a}}^2+1}(\mathit{\_}\mathit{a}f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)-\sqrt{-{\left(f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)\right)}^2+f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)}){\left(f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)\right)}^{-1}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1})x\}$$