3.116   ODE No. 116

$$\boxed{{\displaystyle x\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)-x\sqrt{({\left(y\left(x\right)\right)}^2-x^2)({\left(y\left(x\right)\right)}^2-4x^2)}-y\left(x\right)=0}}$$

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

Mathematica: cpu = 0.410552 (sec), leaf count = 143 $$\text{Solve}[\frac{2{(\frac{y(x)}{x}-2)}^{3/2}\sqrt{-\frac{4}{\frac{y(x)}{x}-2}-1}\sqrt{-\frac{3}{\frac{y(x)}{x}-2}-1}\sqrt{\frac{1}{\frac{y(x)}{x}-2}+1}F({\sin }^{-1}\left(\frac{\sqrt{-1-\frac{3}{\frac{y(x)}{x}-2}}}{\sqrt{2}}\right)|-8)}{\sqrt{\frac{y(x)}{x}-1}\sqrt{\frac{y(x)}{x}+1}\sqrt{\frac{y(x)}{x}+2}}=c_1+\frac{x^2}{2},y(x)]$$

Maple: cpu = 0.156 (sec), leaf count = 144 $$\{{\int }_{\mathit{\_}\mathit{b}}^{x}1(\mathit{\_}\mathit{a}\sqrt{4{\mathit{\_}\mathit{a}}^4-5{\mathit{\_}\mathit{a}}^2{\left(y\left(x\right)\right)}^2+{\left(y\left(x\right)\right)}^4}+y\left(x\right))\frac{1}{\sqrt{4{\mathit{\_}\mathit{a}}^4-5{\mathit{\_}\mathit{a}}^2{\left(y\left(x\right)\right)}^2+{\left(y\left(x\right)\right)}^4}}\mathrm{d}\mathit{\_}\mathit{a}+{\int }^{y\left(x\right)}-1({\int }_{\mathit{\_}\mathit{b}}^x(4{\mathit{\_}\mathit{a}}^4-{\mathit{\_}\mathit{f}}^4){(4{\mathit{\_}\mathit{a}}^4-5{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4)}^{-\frac{3}{2}}\mathrm{d}\mathit{\_}\mathit{a}\sqrt{{\mathit{\_}\mathit{f}}^4-5{\mathit{\_}\mathit{f}}^2{x}^2+4x^4}+x)\frac{1}{\sqrt{{\mathit{\_}\mathit{f}}^4-5{\mathit{\_}\mathit{f}}^2{x}^2+4x^4}}d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$