ODE No. 512

3.512 ODE No. 512

$(a {({(y (x))}^{2} + x^{2})}^{3 / 2} - x^{2}) {(\frac{d}{d x} y (x))}^{2} + 2 x y (x) \frac{d}{d x} y (x) + a {({(y (x))}^{2} + x^{2})}^{3 / 2} - {(y (x))}^{2} = 0$

Mathematica: cpu = 4.686095 (sec), leaf count = 725 ${Solve [\tan^{- 1} (\frac{x}{y (x)}) - \frac{i \sqrt{a} (x^{2} + y (x)^{2}) \sqrt{\sqrt{x^{2} + y (x)^{2}} - a (x^{2} + y (x)^{2})} (\sqrt{2} (\log (\frac{a^{3 / 2} (3 i \sqrt{2} a \sqrt{x^{2} + y (x)^{2}} + 4 \sqrt{a} \sqrt{\sqrt{x^{2} + y (x)^{2}} - a (x^{2} + y (x)^{2})} - i \sqrt{2})}{4 a \sqrt{x^{2} + y (x)^{2}} + 4}) - \log (\frac{- 3 i \sqrt{2} a^{3 / 2} \sqrt{x^{2} + y (x)^{2}} - 4 a \sqrt{\sqrt{x^{2} + y (x)^{2}} - a (x^{2} + y (x)^{2})} + i \sqrt{2} \sqrt{a}}{4 a \sqrt{x^{2} + y (x)^{2}} + 4})) + 2 \log (\frac{- 2 i a \sqrt{x^{2} + y (x)^{2}} + 2 \sqrt{a} \sqrt{\sqrt{x^{2} + y (x)^{2}} - a (x^{2} + y (x)^{2})} + i}{\sqrt{a}}))}{2 \sqrt{a {(x^{2} + y (x)^{2})}^{2} (\sqrt{x^{2} + y (x)^{2}} - a (x^{2} + y (x)^{2}))}} = c_{1}, y (x)], Solve [\tan^{- 1} (\frac{x}{y (x)}) + \frac{i \sqrt{a} (x^{2} + y (x)^{2}) \sqrt{\sqrt{x^{2} + y (x)^{2}} - a (x^{2} + y (x)^{2})} (\sqrt{2} (\log (\frac{a^{3 / 2} (3 i \sqrt{2} a \sqrt{x^{2} + y (x)^{2}} + 4 \sqrt{a} \sqrt{\sqrt{x^{2} + y (x)^{2}} - a (x^{2} + y (x)^{2})} - i \sqrt{2})}{4 a \sqrt{x^{2} + y (x)^{2}} + 4}) - \log (\frac{- 3 i \sqrt{2} a^{3 / 2} \sqrt{x^{2} + y (x)^{2}} - 4 a \sqrt{\sqrt{x^{2} + y (x)^{2}} - a (x^{2} + y (x)^{2})} + i \sqrt{2} \sqrt{a}}{4 a \sqrt{x^{2} + y (x)^{2}} + 4})) + 2 \log (\frac{- 2 i a \sqrt{x^{2} + y (x)^{2}} + 2 \sqrt{a} \sqrt{\sqrt{x^{2} + y (x)^{2}} - a (x^{2} + y (x)^{2})} + i}{\sqrt{a}}))}{2 \sqrt{a {(x^{2} + y (x)^{2})}^{2} (\sqrt{x^{2} + y (x)^{2}} - a (x^{2} + y (x)^{2}))}} = c_{1}, y (x)]}$

Maple: cpu = 40.935 (sec), leaf count = 135 ${y (x) = x {(\tan (R o o t O f (-_Z + \int^{\frac{x^{2} ({(\tan (_Z))}^{2} + 1)}{{(\tan (_Z))}^{2}}} - \frac{1}{2 {_a}^{2} (_a a^{2} - 1)} (\sqrt{_a} a + 1) \sqrt{- {_a}^{\frac{5}{2}} a (\sqrt{_a} a - 1)} d_a +_C 1)))}^{- 1}, y (x) = x {(\tan (R o o t O f (-_Z + \int^{\frac{x^{2} ({(\tan (_Z))}^{2} + 1)}{{(\tan (_Z))}^{2}}} \frac{1}{2 {_a}^{2} (_a a^{2} - 1)} (\sqrt{_a} a + 1) \sqrt{- {_a}^{\frac{5}{2}} a (\sqrt{_a} a - 1)} d_a +_C 1)))}^{- 1}}$

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