ODE No. 493

2.493 ODE No. 493

$(a^{2} - 2 a x + y (x)^{2}) y^{'} (x)^{2} + 2 a y (x) y^{'} (x) + y (x)^{2} = 0$ ✓ Mathematica : cpu = 8.45034 (sec), leaf count = 553

${Solve [{y (x) = \frac{- \sqrt{- a {K $ 1788602}^{2} (a {K $ 1788602}^{2} - 2 {K $ 1788602}^{2} x - 2 x)} - a K $ 1788602}{{K $ 1788602}^{2} + 1}, x = \frac{a c_{1}^{2} {K $ 1788602}^{2} - 2 a c_{1} \sqrt{{K $ 1788602}^{2} + 1} - 2 a c_{1} {K $ 1788602}^{2} \log (K $ 1788602) + 2 a c_{1} {K $ 1788602}^{2} \log (\sqrt{{K $ 1788602}^{2} + 1} + 1) + 2 a c_{1} \log (\sqrt{{K $ 1788602}^{2} + 1} + 1) - 2 a c_{1} \log (K $ 1788602) + a c_{1}^{2} + a {K $ 1788602}^{2} + a {K $ 1788602}^{2} \log^{2} (K $ 1788602) + a {K $ 1788602}^{2} \log^{2} (\sqrt{{K $ 1788602}^{2} + 1} + 1) + a \log^{2} (\sqrt{{K $ 1788602}^{2} + 1} + 1) - 2 a {K $ 1788602}^{2} \log (K $ 1788602) \log (\sqrt{{K $ 1788602}^{2} + 1} + 1) + 2 a \sqrt{{K $ 1788602}^{2} + 1} \log (K $ 1788602) - 2 a \log (K $ 1788602) \log (\sqrt{{K $ 1788602}^{2} + 1} + 1) - 2 a \sqrt{{K $ 1788602}^{2} + 1} \log (\sqrt{{K $ 1788602}^{2} + 1} + 1) + a \log^{2} (K $ 1788602) + a}{2 ({K $ 1788602}^{2} + 1)}}, {y (x), K $ 1788602}], Solve [{y (x) = \frac{\sqrt{- a {K $ 1788607}^{2} (a {K $ 1788607}^{2} - 2 {K $ 1788607}^{2} x - 2 x)} - a K $ 1788607}{{K $ 1788607}^{2} + 1}, x = \frac{a c_{1}^{2} {K $ 1788607}^{2} - 2 a c_{1} \sqrt{{K $ 1788607}^{2} + 1} - 2 a c_{1} {K $ 1788607}^{2} \log (K $ 1788607) + 2 a c_{1} {K $ 1788607}^{2} \log (\sqrt{{K $ 1788607}^{2} + 1} + 1) + 2 a c_{1} \log (\sqrt{{K $ 1788607}^{2} + 1} + 1) - 2 a c_{1} \log (K $ 1788607) + a c_{1}^{2} + a {K $ 1788607}^{2} + a {K $ 1788607}^{2} \log^{2} (K $ 1788607) + a {K $ 1788607}^{2} \log^{2} (\sqrt{{K $ 1788607}^{2} + 1} + 1) + a \log^{2} (\sqrt{{K $ 1788607}^{2} + 1} + 1) - 2 a {K $ 1788607}^{2} \log (K $ 1788607) \log (\sqrt{{K $ 1788607}^{2} + 1} + 1) + 2 a \sqrt{{K $ 1788607}^{2} + 1} \log (K $ 1788607) - 2 a \log (K $ 1788607) \log (\sqrt{{K $ 1788607}^{2} + 1} + 1) - 2 a \sqrt{{K $ 1788607}^{2} + 1} \log (\sqrt{{K $ 1788607}^{2} + 1} + 1) + a \log^{2} (K $ 1788607) + a}{2 ({K $ 1788607}^{2} + 1)}}, {y (x), K $ 1788607}]}$

✓ Maple : cpu = 1.115 (sec), leaf count = 124

${[x (_T) = \frac{1}{2 a} ({(A r t a n h (\frac{1}{\sqrt{{_T}^{2} + 1}}))}^{2} \sqrt{{_T}^{2} + 1} a^{2} - 2 A r t a n h (\frac{1}{\sqrt{{_T}^{2} + 1}}) \sqrt{{_T}^{2} + 1}_C 1 a - 2 A r t a n h (\frac{1}{\sqrt{{_T}^{2} + 1}}) a^{2} + \sqrt{{_T}^{2} + 1} {_C 1}^{2} + a^{2} \sqrt{{_T}^{2} + 1} + 2_C 1 a) \frac{1}{\sqrt{{_T}^{2} + 1}}, y (_T) = -_T (a A r t a n h (\frac{1}{\sqrt{{_T}^{2} + 1}}) -_C 1) \frac{1}{\sqrt{{_T}^{2} + 1}}]}$

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