3.516   ODE No. 516

$$\boxed{{\displaystyle ({\left(y\left(x\right)\right)}^2+x^2)f\left(\frac{x}{\sqrt{{\left(y\left(x\right)\right)}^2+x^2}}\right)({\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^2+1)-{(x\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)-y\left(x\right))}^2=0}}$$

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

Mathematica: cpu = 2.612332 (sec), leaf count = 251 $$\{\text{Solve}[{\int }_1^{\frac{y(x)}{x}}\frac{K[1{]}^{2}f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)+f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)-1}{(K[1]-i)(K[1]+i)\sqrt{f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)}(K[1]\sqrt{f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)}+i\sqrt{f\left(\frac{1}{\sqrt{K[1{]}^2+1}}\right)-1})}dK[1]=c_1-\log (x),y(x)],\text{Solve}[{\int }_1^{\frac{y(x)}{x}}\frac{K[2{]}^{2}f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)+f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)-1}{(K[2]-i)(K[2]+i)\sqrt{f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)}(K[2]\sqrt{f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)}-i\sqrt{f\left(\frac{1}{\sqrt{K[2{]}^2+1}}\right)-1})}dK[2]=c_1-\log (x),y(x)]\}$$

Maple: cpu = 0.827 (sec), leaf count = 141 $$\{y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-\ln \left(x\right)+{\int }^{\mathit{\_}\mathit{Z}}-\frac{1}{{\mathit{\_}\mathit{a}}^2+1}(\mathit{\_}\mathit{a}f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)-\sqrt{-{\left(f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)\right)}^2+f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)}){\left(f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)\right)}^{-1}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1})x,y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-\ln \left(x\right)+{\int }^{\mathit{\_}\mathit{Z}}-\frac{1}{{\mathit{\_}\mathit{a}}^2+1}(\mathit{\_}\mathit{a}f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)+\sqrt{-{\left(f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)\right)}^2+f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)}){\left(f\left(\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}\right)\right)}^{-1}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1})x\}$$