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

Question Number 60695 by maxmathsup by imad last updated on 24/May/19

s o l v e \sqrt{3 + x^{2}} y^{^{″}} - (2 x + 1) y^{^{'}} = x^{2} e^{- x^{2}}

Commented by maxmathsup by imad last updated on 27/May/19

l e t y^{^{'}} = z s o (e) \Leftrightarrow \sqrt{3 + x^{2}} z^{^{'}} - (2 x + 1) z = x^{2} e^{- x^{2}}

(h e) \to \sqrt{3 + x^{2}} z^{^{'}} - (2 x + 1) z = 0 \Rightarrow \sqrt{3 + x^{2}} z^{^{'}} = (2 x + 1) z \Rightarrow \frac{z^{^{'}}}{z} = \frac{2 x + 1}{\sqrt{x^{2} + 3}} \Rightarrow

l n ∣ z ∣ = \int \frac{2 x + 1}{\sqrt{x^{2} + 3}} d x + c = 2 \int \frac{x}{\sqrt{x^{2} + 3}} + \int \frac{d x}{\sqrt{x^{2} + 3}} + c

= 2 \sqrt{x^{2} + 3} + \int \frac{d x}{\sqrt{x^{2} + 3}} d x

\int \frac{d x}{\sqrt{x^{2} + 3}} d x =_{x = \sqrt{3} u} \int \frac{\sqrt{3} d u}{\sqrt{3} \sqrt{1 + u^{2}}} = l n (u + \sqrt{1 + u^{2}}) + c_{0}

= l n (\frac{x}{\sqrt{3}} + \sqrt{1 + \frac{x^{2}}{3}}) + c_{0} \Rightarrow

l n ∣ z ∣ = 2 \sqrt{x^{2} + 3} + l n (\frac{x}{\sqrt{3}} + \sqrt{1 + \frac{x^{2}}{3}}) + c \Rightarrow

z (x) = k (\frac{x}{\sqrt{3}} + \sqrt{1 + \frac{x^{2}}{3}}) e^{2 \sqrt{x^{2} + 3}} m v c m e t h o d g i v e

z^{^{'}} (x) = K^{^{'}} (\frac{x}{\sqrt{3}} + \sqrt{1 + \frac{x^{2}}{3}}) e^{2 \sqrt{1 + \frac{x^{2}}{3}}} + K {(\frac{1}{\sqrt{3}} + \frac{\frac{2 x}{\sqrt{3}}}{2 \sqrt{1 + \frac{x^{2}}{3}}}) e^{2 \sqrt{1 + \frac{x^{2}}{3}}}

+ (\frac{x}{\sqrt{3}} + \sqrt{1 + \frac{x^{2}}{3}}) (2 \frac{2 x}{3}) \frac{1}{\sqrt{1 + \frac{x^{2}}{3}}} e^{2 \sqrt{1 + \frac{x^{2}}{3}}}}

= K^{^{'}} (\frac{x + \sqrt{3 + x^{2}}}{\sqrt{3}}) e^{\frac{2}{\sqrt{3}} \sqrt{3 + x^{2}}} + K {\frac{1}{\sqrt{3}} + \frac{1}{\sqrt{3 + x^{2}}}} e^{\frac{2}{\sqrt{3}} \sqrt{3 + x^{2}}}

+ \frac{4 x}{3} (\frac{x + \sqrt{3 + x^{2}}}{\sqrt{3}}) \frac{1}{\sqrt{3 + x^{2}}} e^{\frac{2}{\sqrt{3}} \sqrt{3 + x^{2}}} \dots . b e c o n t i n u e d \dots .