ODE No. 406

3.406 ODE No. 406

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

Mathematica: cpu = 0.864110 (sec), leaf count = 49 $Solve [{x = \frac{a K $ 1211116 \sinh^{- 1} (K $ 1211116)}{\sqrt{{K $ 1211116}^{2} + 1}} + \frac{c_{1} K $ 1211116}{\sqrt{{K $ 1211116}^{2} + 1}}, y (x) = a K $ 1211116 - \frac{x}{K $ 1211116}}, {y (x), K $ 1211116}]$

Maple: cpu = 0.577 (sec), leaf count = 266 ${_C 1 (- y (x) + \sqrt{4 a x + {(y (x))}^{2}}) \frac{1}{\sqrt{- \frac{1}{a^{2}} (2 y (x) \sqrt{4 a x + {(y (x))}^{2}} - 4 a^{2} - 4 a x - 2 {(y (x))}^{2})}} + x - 1 (- y (x) + \sqrt{4 a x + {(y (x))}^{2}}) A r c s i n h (\frac{1}{2 a} (- y (x) + \sqrt{4 a x + {(y (x))}^{2}})) \frac{1}{\sqrt{- \frac{1}{a^{2}} (2 y (x) \sqrt{4 a x + {(y (x))}^{2}} - 4 a^{2} - 4 a x - 2 {(y (x))}^{2})}} = 0,_C 1 (y (x) + \sqrt{4 a x + {(y (x))}^{2}}) \frac{1}{\sqrt{\frac{1}{a^{2}} (y (x) \sqrt{4 a x + {(y (x))}^{2}} + 2 a^{2} + 2 a x + {(y (x))}^{2})}} + x - \frac{\sqrt{2}}{2} (y (x) + \sqrt{4 a x + {(y (x))}^{2}}) A r c s i n h (\frac{1}{2 a} (y (x) + \sqrt{4 a x + {(y (x))}^{2}})) \frac{1}{\sqrt{\frac{1}{a^{2}} (y (x) \sqrt{4 a x + {(y (x))}^{2}} + 2 a^{2} + 2 a x + {(y (x))}^{2})}} = 0}$

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