Resolve-y-xy-a-1-y-2-

Question Number 168742 by LEKOUMA last updated on 16/Apr/22

R e s o l v e

y = {x y}^{'} + a \sqrt{1 + {(y^{'})}^{2}}

Answered by mindispower last updated on 19/Apr/22

\Rightarrow

y^{'} = {x y}^{″} + y^{'} + a \frac{2 y^{'} y^{″}}{2 \sqrt{1 + y^{' 2}}}

y^{'} = z

{x z}^{'} + \frac{{z z}^{'}}{\sqrt{1 + z^{2}}} = 0

z^{'} (x + a \frac{z}{\sqrt{1 + z^{2}}}) = 0

b y c o n t i n u i t y \Leftrightarrow {\begin{cases} z^{'} = 0 \\ \frac{z}{\sqrt{1 + z^{2}}} = - \frac{1}{a} x \end{cases}

z = c

o r z^{2} = x^{2} (1 + z^{2}) \Rightarrow z^{2} = \frac{{(\frac{x}{a})}^{2}}{1 - {(\frac{x}{a})}^{2}}, x \in] - 1, 1 [

z = \frac{- x}{\sqrt{a^{2} - x^{2}}}, \Rightarrow y = \sqrt{a^{2} - x^{2}} + c, a > 0

y = \frac{- x^{2}}{\sqrt{a^{2} - x^{2}}} + a \sqrt{1 + \frac{x^{2}}{a^{2} - x^{2}}} \Rightarrow c = 0

c x + d = x (c) + a \sqrt{1 + c^{2}}

d = a \sqrt{1 + c^{2}}

S = {c x + a \sqrt{1 + c^{2}}, \sqrt{a^{2} - x^{2}}}