ODE No. 493

2.493 ODE No. 493

Problem in Latex
Mathematica input
Maple input

$(a^{2} - 2 a x + y (x)^{2}) y^{'} (x)^{2} + 2 a y (x) y^{'} (x) + y (x)^{2} = 0$ ✓ Mathematica : cpu = 7.08136 (sec), leaf count = 553

${Solve [{y (x) = \frac{- \sqrt{- a {K $ 4089}^{2} (a {K $ 4089}^{2} - 2 {K $ 4089}^{2} x - 2 x)} - a K $ 4089}{{K $ 4089}^{2} + 1}, x = \frac{a {K $ 4089}^{2} + a {K $ 4089}^{2} \log^{2} (K $ 4089) + a {K $ 4089}^{2} \log^{2} (\sqrt{{K $ 4089}^{2} + 1} + 1) + a \log^{2} (\sqrt{{K $ 4089}^{2} + 1} + 1) - 2 a {K $ 4089}^{2} \log (K $ 4089) \log (\sqrt{{K $ 4089}^{2} + 1} + 1) + 2 a \sqrt{{K $ 4089}^{2} + 1} \log (K $ 4089) - 2 a \log (K $ 4089) \log (\sqrt{{K $ 4089}^{2} + 1} + 1) - 2 a \sqrt{{K $ 4089}^{2} + 1} \log (\sqrt{{K $ 4089}^{2} + 1} + 1) + a c_{1}^{2} {K $ 4089}^{2} - 2 a c_{1} \sqrt{{K $ 4089}^{2} + 1} - 2 a c_{1} {K $ 4089}^{2} \log (K $ 4089) + 2 a c_{1} {K $ 4089}^{2} \log (\sqrt{{K $ 4089}^{2} + 1} + 1) + 2 a c_{1} \log (\sqrt{{K $ 4089}^{2} + 1} + 1) + a \log^{2} (K $ 4089) - 2 a c_{1} \log (K $ 4089) + a + a c_{1}^{2}}{2 ({K $ 4089}^{2} + 1)}}, {y (x), K $ 4089}], Solve [{y (x) = \frac{\sqrt{- a {K $ 4644}^{2} (a {K $ 4644}^{2} - 2 {K $ 4644}^{2} x - 2 x)} - a K $ 4644}{{K $ 4644}^{2} + 1}, x = \frac{a {K $ 4644}^{2} + a {K $ 4644}^{2} \log^{2} (K $ 4644) + a {K $ 4644}^{2} \log^{2} (\sqrt{{K $ 4644}^{2} + 1} + 1) + a \log^{2} (\sqrt{{K $ 4644}^{2} + 1} + 1) - 2 a {K $ 4644}^{2} \log (K $ 4644) \log (\sqrt{{K $ 4644}^{2} + 1} + 1) + 2 a \sqrt{{K $ 4644}^{2} + 1} \log (K $ 4644) - 2 a \log (K $ 4644) \log (\sqrt{{K $ 4644}^{2} + 1} + 1) - 2 a \sqrt{{K $ 4644}^{2} + 1} \log (\sqrt{{K $ 4644}^{2} + 1} + 1) + a c_{1}^{2} {K $ 4644}^{2} - 2 a c_{1} \sqrt{{K $ 4644}^{2} + 1} - 2 a c_{1} {K $ 4644}^{2} \log (K $ 4644) + 2 a c_{1} {K $ 4644}^{2} \log (\sqrt{{K $ 4644}^{2} + 1} + 1) + 2 a c_{1} \log (\sqrt{{K $ 4644}^{2} + 1} + 1) + a \log^{2} (K $ 4644) - 2 a c_{1} \log (K $ 4644) + a + a c_{1}^{2}}{2 ({K $ 4644}^{2} + 1)}}, {y (x), K $ 4644}]}$ ✓ Maple : cpu = 1.255 (sec), leaf count = 111

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

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