ODE No. 406

2.406 ODE No. 406

$a y^{'} (x)^{2} - y (x) y^{'} (x) - x = 0$ ✓ Mathematica : cpu = 0.805152 (sec), leaf count = 49

$Solve [{x = \frac{a K $ 1239384 \sinh^{- 1} (K $ 1239384)}{\sqrt{{K $ 1239384}^{2} + 1}} + \frac{c_{1} K $ 1239384}{\sqrt{{K $ 1239384}^{2} + 1}}, y (x) = a K $ 1239384 - \frac{x}{K $ 1239384}}, {y (x), K $ 1239384}]$

✓ Maple : cpu = 0.081 (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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