ODE No. 402

2.402 ODE No. 402

Problem in Latex
Mathematica input
Maple input

$x^{2} + 4 x y^{'} (x) + 3 y^{'} (x)^{2} - y (x) = 0$ ✓ Mathematica : cpu = 6.5522 (sec), leaf count = 1211

${Solve [\frac{1}{2} (\frac{\tanh^{- 1} (\frac{1 - 3 x}{2 \sqrt{1 - 9 y (x)}})}{\sqrt{1 - 9 y (x)}} + \log (- 3 x^{2} + 2 x - 12 y (x) + 1) + \frac{9 \tanh^{- 1} (\frac{- 2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}}) y (x)}{\sqrt{1 - 9 y (x)} \sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5}} - \frac{9 \tanh^{- 1} (\frac{2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}}) y (x)}{\sqrt{1 - 9 y (x)} \sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5}} + \frac{\tanh^{- 1} (\frac{3 x - 1}{2 \sqrt{1 - 9 y (x)}})}{\sqrt{1 - 9 y (x)}} + \frac{2 \tanh^{- 1} (\frac{- 2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}})}{\sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5}} - \frac{\tanh^{- 1} (\frac{- 2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}})}{\sqrt{1 - 9 y (x)} \sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5}} + \frac{2 \tanh^{- 1} (\frac{2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}})}{\sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5}} + \frac{\tanh^{- 1} (\frac{2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}})}{\sqrt{1 - 9 y (x)} \sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5}}) = c_{1}, y (x)], Solve [\frac{1}{2} (\frac{\tanh^{- 1} (\frac{1 - 3 x}{2 \sqrt{1 - 9 y (x)}})}{\sqrt{1 - 9 y (x)}} + \log (- 3 x^{2} + 2 x - 12 y (x) + 1) - \frac{9 \tanh^{- 1} (\frac{- 2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}}) y (x)}{\sqrt{1 - 9 y (x)} \sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5}} + \frac{9 \tanh^{- 1} (\frac{2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}}) y (x)}{\sqrt{1 - 9 y (x)} \sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5}} + \frac{\tanh^{- 1} (\frac{3 x - 1}{2 \sqrt{1 - 9 y (x)}})}{\sqrt{1 - 9 y (x)}} - \frac{2 \tanh^{- 1} (\frac{- 2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}})}{\sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5}} + \frac{\tanh^{- 1} (\frac{- 2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}})}{\sqrt{1 - 9 y (x)} \sqrt{- 9 y (x) - 4 \sqrt{1 - 9 y (x)} + 5}} - \frac{2 \tanh^{- 1} (\frac{2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}})}{\sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5}} - \frac{\tanh^{- 1} (\frac{2 \sqrt{1 - 9 y (x)} x + x + 9 y (x)}{\sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5} \sqrt{x^{2} + 3 y (x)}})}{\sqrt{1 - 9 y (x)} \sqrt{- 9 y (x) + 4 \sqrt{1 - 9 y (x)} + 5}}) = c_{1}, y (x)]}$ ✓ Maple : cpu = 0.154 (sec), leaf count = 101

${y (x) = - \frac{x^{2}}{3}, y (x) = \frac{- 3 c_{1}^{2} x^{2} - 2 c_{1} \sqrt{3} x + 3}{12 c_{1}^{2}}, y (x) = \frac{- 3 c_{1}^{2} x^{2} + 2 c_{1} \sqrt{3} x + 3}{12 c_{1}^{2}}, y (x) = \frac{c_{1}^{2}}{4} - \frac{c_{1} \sqrt{3} x}{6} - \frac{x^{2}}{4}, y (x) = \frac{c_{1}^{2}}{4} + \frac{c_{1} \sqrt{3} x}{6} - \frac{x^{2}}{4}}$

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