ODE No. 405

3.405 ODE No. 405

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

Mathematica: cpu = 1.013629 (sec), leaf count = 53 $Solve [{x = \frac{a K $ 1210941 \sin^{- 1} (K $ 1210941)}{\sqrt{1 - {K $ 1210941}^{2}}} + \frac{c_{1} K $ 1210941}{\sqrt{1 - {K $ 1210941}^{2}}}, y (x) = \frac{x}{K $ 1210941} - a K $ 1210941}, {y (x), K $ 1210941}]$

Maple: cpu = 0.593 (sec), leaf count = 380 ${-_C 1 (- y (x) + \sqrt{4 a x + {(y (x))}^{2}}) \frac{1}{\sqrt{\frac{1}{a} (- y (x) + \sqrt{4 a x + {(y (x))}^{2}} + 2 a)}} \frac{1}{\sqrt{\frac{1}{a} (- y (x) + \sqrt{4 a x + {(y (x))}^{2}} - 2 a)}} + x + 1 (- y (x) + \sqrt{4 a x + {(y (x))}^{2}}) \ln (\frac{1}{2 a} (\sqrt{\frac{1}{a^{2}} (4 a x + 2 {(y (x))}^{2} - 2 y (x) \sqrt{4 a x + {(y (x))}^{2}} - 4 a^{2})} a + \sqrt{4 a x + {(y (x))}^{2}} - y (x))) \frac{1}{\sqrt{- \frac{1}{a^{2}} (2 y (x) \sqrt{4 a x + {(y (x))}^{2}} + 4 a^{2} - 4 a x - 2 {(y (x))}^{2})}} = 0,_C 1 (y (x) + \sqrt{4 a x + {(y (x))}^{2}}) \frac{1}{\sqrt{\frac{1}{a} (- 2 y (x) - 2 \sqrt{4 a x + {(y (x))}^{2}} + 4 a)}} \frac{1}{\sqrt{\frac{1}{a} (- 2 y (x) - 2 \sqrt{4 a x + {(y (x))}^{2}} - 4 a)}} + x - \frac{\sqrt{2}}{2} (y (x) + \sqrt{4 a x + {(y (x))}^{2}}) \ln (\frac{1}{2 a} (\sqrt{2} \sqrt{\frac{1}{a^{2}} (y (x) \sqrt{4 a x + {(y (x))}^{2}} - 2 a^{2} + 2 a x + {(y (x))}^{2})} a - \sqrt{4 a x + {(y (x))}^{2}} - y (x))) \frac{1}{\sqrt{\frac{1}{a^{2}} (y (x) \sqrt{4 a x + {(y (x))}^{2}} - 2 a^{2} + 2 a x + {(y (x))}^{2})}} = 0}$

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