3.511   ODE No. 511

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

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

Mathematica: cpu = 1.555697 (sec), leaf count = 229 $$\{\text{Solve}[{\tan }^{-1}\left(\frac{x}{y(x)}\right)-\frac{2\sqrt{a^2(x^2+y(x{)}^2)(\sqrt{x^2+y(x{)}^2}-a^2)}{\tan }^{-1}\left(\frac{\sqrt{\sqrt{x^2+y(x{)}^2}-a^2}}{a}\right)}{a\sqrt{x^2+y(x{)}^2}\sqrt{\sqrt{x^2+y(x{)}^2}-a^2}}=c_1,y(x)],\text{Solve}[\frac{2\sqrt{a^2(x^2+y(x{)}^2)(\sqrt{x^2+y(x{)}^2}-a^2)}{\tan }^{-1}\left(\frac{\sqrt{\sqrt{x^2+y(x{)}^2}-a^2}}{a}\right)}{a\sqrt{x^2+y(x{)}^2}\sqrt{\sqrt{x^2+y(x{)}^2}-a^2}}+{\tan }^{-1}\left(\frac{x}{y(x)}\right)=c_1,y(x)]\}$$

Maple: cpu = 8.112 (sec), leaf count = 199 $$\{\arctan \left(\frac{x}{y\left(x\right)}\right)-2\frac{\sqrt{a^2({\left(y\left(x\right)\right)}^2+x^2)(-a^2+\sqrt{{\left(y\left(x\right)\right)}^2+x^2})}}{a\sqrt{{\left(y\left(x\right)\right)}^2+x^2}\sqrt{-a^2+\sqrt{{\left(y\left(x\right)\right)}^2+x^2}}}\arctan \left(\frac{\sqrt{-a^2+\sqrt{{\left(y\left(x\right)\right)}^2+x^2}}}{a}\right)-\mathit{\_}\mathit{C}\mathit{1}=0,\arctan \left(\frac{x}{y\left(x\right)}\right)+2\frac{\sqrt{a^2({\left(y\left(x\right)\right)}^2+x^2)(-a^2+\sqrt{{\left(y\left(x\right)\right)}^2+x^2})}}{a\sqrt{{\left(y\left(x\right)\right)}^2+x^2}\sqrt{-a^2+\sqrt{{\left(y\left(x\right)\right)}^2+x^2}}}\arctan \left(\frac{\sqrt{-a^2+\sqrt{{\left(y\left(x\right)\right)}^2+x^2}}}{a}\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$