3.558   ODE No. 558

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

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

Mathematica: cpu = 0.586574 (sec), leaf count = 395 $$\{\text{Solve}[\frac{a(-\log \left(\frac{(a^2-1)(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2-\frac{iy(x)}{x}-1)}{a^3(\frac{y(x)}{x}-i)}\right)+\log (-\frac{(a^2-1)(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2+\frac{iy(x)}{x}-1)}{a^3(\frac{y(x)}{x}+i)})+\log (\frac{y(x{)}^2}{x^2}+1))-2i{\tan }^{-1}\left(\frac{y(x)}{x\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}}\right)}{2(a^2-1)}=\frac{a\log (x-a^{2}x)}{1-a^2}+c_1,y(x)],\text{Solve}[\frac{2i{\tan }^{-1}\left(\frac{y(x)}{x\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}}\right)+a(\log (-\frac{(a^2-1)(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2-\frac{iy(x)}{x}-1)}{a^3(\frac{y(x)}{x}-i)})-\log \left(\frac{(a^2-1)(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2+\frac{iy(x)}{x}-1)}{a^3(\frac{y(x)}{x}+i)}\right)+\log (\frac{y(x{)}^2}{x^2}+1))}{2(a^2-1)}=\frac{a\log (x-a^{2}x)}{1-a^2}+c_1,y(x)]\}$$

Maple: cpu = 0.140 (sec), leaf count = 223 $$\{x-\mathit{\_}\mathit{C}\mathit{1}{\mathrm{e}}^{\frac{1}{a}\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(\frac{1}{(a^2-1)x}(\sqrt{-a^2{x}^2+x^2+{\left(y\left(x\right)\right)}^2}a+y\left(x\right))\right)}\frac{1}{\sqrt{\frac{1}{{(a^2-1)}^2{x}^2}(-a^2{x}^2+a^2{\left(y\left(x\right)\right)}^2+2\sqrt{-a^2{x}^2+x^2+{\left(y\left(x\right)\right)}^2}ay\left(x\right)+x^2+{\left(y\left(x\right)\right)}^2)}}=0,x-\mathit{\_}\mathit{C}\mathit{1}{\mathrm{e}}^{-\frac{1}{a}\mathit{A}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(\frac{1}{(a^2-1)x}(\sqrt{-a^2{x}^2+x^2+{\left(y\left(x\right)\right)}^2}a-y\left(x\right))\right)}\frac{1}{\sqrt{-\frac{1}{{(a^2-1)}^2{x}^2}(a^2{x}^2-a^2{\left(y\left(x\right)\right)}^2+2\sqrt{-a^2{x}^2+x^2+{\left(y\left(x\right)\right)}^2}ay\left(x\right)-x^2-{\left(y\left(x\right)\right)}^2)}}=0\}$$