solve-x-1-y-x-2-y-x-2-e-2x-with-y-3-1-

Question Number 57820 by maxmathsup by imad last updated on 12/Apr/19

s o l v e \sqrt{x + 1} y^{^{'}} - \sqrt{x - 2} y = x^{2} e^{- 2 x} w i t h y (3) = 1

Commented by maxmathsup by imad last updated on 14/Apr/19

(h e) \Rightarrow \sqrt{x + 1} y^{^{'}} - \sqrt{x - 2} y = 0 \Rightarrow \sqrt{x + 1} y^{^{'}} = \sqrt{x - 2} y \Rightarrow \frac{y^{^{'}}}{y} = \frac{\sqrt{x - 2}}{\sqrt{x + 1}} \Rightarrow

l n ∣ y ∣ = \int \sqrt{\frac{x - 2}{x + 1}} d x + c c h a n g e m e n t \sqrt{\frac{x - 2}{x + 1}} = t g i v e \frac{x - 2}{x + 1} = t^{2} \Rightarrow

x - 2 = t^{2} x + t^{2} \Rightarrow (1 - t^{2}) x = t^{2} + 2 \Rightarrow x = \frac{t^{2} + 2}{1 - t^{2}} \Rightarrow \frac{d x}{d t} = \frac{2 t (1 - t^{2}) - (t^{2} + 2) (- 2 t)}{{(1 - t^{2})}^{2}}

= \frac{2 t - 2 t^{3} + 2 t^{3} + 4 t}{{(1 - t^{2})}^{2}} = \frac{6 t}{{(1 - t^{2})}^{2}} \Rightarrow \int \sqrt{\frac{x - 2}{x + 1}} d x = \int \frac{6 t^{2}}{{(1 - t^{2})}^{2}} d t

w e h a v e \int \frac{t^{2}}{{(t^{2} - 1)}^{2}} d t = \int t \frac{t}{{(t^{2} - 1)}^{2}} d t b y p a r t s u = t a n d v^{^{'}} = \frac{t}{{(t^{2} - 1)}^{2}}

\Rightarrow \int \frac{t^{2}}{{(t^{2} - 1)}^{2}} d t = - \frac{1}{2 (t^{2} - 1)} t - \int - \frac{1}{2 (t^{2} - 1)} d t

= - \frac{t}{2 (t^{2} - 1)} + \frac{1}{2} \int (\frac{1}{t - 1} - \frac{1}{t + 1}) d t = - \frac{t}{2 (t^{2} - 1)} + \frac{1}{2} l n ∣ \frac{t - 1}{t + 1} ∣ \Rightarrow

\int \sqrt{\frac{x - 2}{x + 1}} d x = - \frac{\sqrt{\frac{x - 2}{x + 1}}}{2 (\frac{x - 2}{x + 1} - 1)} + \frac{1}{2} l n ∣ \frac{\sqrt{\frac{x - 2}{x + 1}} - 1}{\sqrt{\frac{x - 2}{x + 1}} + 1} ∣

= - \frac{\sqrt{x - 2}}{2 \sqrt{x + 1}} \frac{1}{(\frac{- 3}{x + 1})} + \frac{1}{2} l n ∣ \frac{\sqrt{x - 2} - \sqrt{x + 1}}{\sqrt{x - 2} + \sqrt{x + 1}} ∣ = \frac{\sqrt{x - 2} (x + 1)}{6 \sqrt{x + 1}} + \frac{1}{2} l n (\dots .)

= \frac{\sqrt{x - 2} \sqrt{x + 1}}{6} + \frac{1}{2} l n ∣ \frac{\sqrt{x - 2} - \sqrt{x + 1}}{\sqrt{x - 2} + \sqrt{x + 1}} ∣ \Rightarrow

y (x) = K \sqrt{\frac{\sqrt{x + 1} - \sqrt{x - 2}}{\sqrt{x + 1} + \sqrt{x - 2}}} e^{\frac{\sqrt{x - 2} \sqrt{x + 1}}{6}} a f t e r w e u s e t h e m v c m e t h o d \dots

b e c o n t i n u e d \dots