solve-the-d-e-x-2-1-y-xy-x-2-e-x-

Question Number 28529 by abdo imad last updated on 26/Jan/18

s o l v e t h e d . e . (x^{2} - 1) y^{^{'}} + x y = x^{2} - e^{x} .

Commented by abdo imad last updated on 28/Jan/18

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

\Rightarrow l n ∣ y ∣= \int \frac{x}{1 - x^{2}} d x = \frac{- 1}{2} l n ∣ 1 - x^{2} ∣ + c

= l n (\frac{1}{\sqrt{∣ 1 - x^{2} ∣}}) + c \Rightarrow y = \frac{λ}{\sqrt{∣ 1 - x^{2} ∣}} l e t u s e m v c m e t h o d

c a s e 1 ∣ x ∣< 1 \Rightarrow y = λ {(1 - x^{2})}^{- \frac{1}{2}}

y^{^{'}} = λ^{^{'}} {(1 - x^{2})}^{\frac{- 1}{2}} + λ x {(1 - x^{2})}^{\frac{- 3}{2}} (e) \Rightarrow

- λ^{^{'}} {(1 - x^{2})}^{\frac{1}{2}} - λ x {(1 - x^{2})}^{\frac{- 1}{2}} + λ x {(1 - x^{2})}^{\frac{- 1}{2}} = x^{2} - e^{x}

\Rightarrow λ^{^{'}} {(1 - x^{2})}^{\frac{1}{2}} = e^{x} - x^{2}

\Rightarrow λ^{^{'}} = (e^{x} - x^{2}) {(1 - x^{2})}^{\frac{- 1}{2}} \Rightarrow

λ (x) = \int_{.}^{x} \frac{e^{t} - t^{2}}{\sqrt{1 - t^{2}}} d t + k a n d

y (x) = \frac{1}{\sqrt{1 - x^{2}}} (\int_{.}^{x} \frac{e^{t} - t^{2}}{\sqrt{1 - t^{2}}} d t + k) .

Commented by abdo imad last updated on 28/Jan/18

f o r c a s e 2 ∣ x ∣ > 1 w e f o l o w t h e s a m e m e t h o d .