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 = 4.26237 (sec), leaf count = 229

$$\{\text{Solve}[c_1=\log (x)+{\int }_1^{\frac{y(x)}{x}}\frac{(K[1{]}^2+1)f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)-1}{(K[1]-i)(K[1]+i)\sqrt{f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)}(K[1]\sqrt{f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)}+i\sqrt{f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)-1})}dK[1],y(x)],\text{Solve}[c_1=\log (x)+{\int }_1^{\frac{y(x)}{x}}\frac{(K[2{]}^2+1)f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)-1}{(K[2]-i)(K[2]+i)\sqrt{f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)}(K[2]\sqrt{f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)}-i\sqrt{f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)-1})}dK[2],y(x)]\}$$

✓ Maple : cpu = 3.192 (sec), leaf count = 139

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