solve-the-differencial-equation-1-x-2-y-xy-1-

Question Number 27594 by abdo imad last updated on 10/Jan/18

s o l v e t h e d i f f e r e n c i a l e q u a t i o n

(1 - x^{2}) y^{^{'}} - x y = 1 .

Commented by abdo imad last updated on 12/Jan/18

e . h \Rightarrow (1 - x^{2}) y^{^{'}} - x y = 0 \Leftrightarrow (1 - x^{2}) y^{,} = x y

\Leftrightarrow \frac{y^{^{'}}}{y} = \frac{x}{1 - x^{2}} \Leftrightarrow l n / y / = \int \frac{x}{1 - x^{2}} d x + k

l n / y / = - \frac{1}{2} l n / 1 - x^{2} / + k = l n (\frac{1}{\sqrt{/ 1 - x^{2}}}) + k

\Rightarrow y = \frac{λ}{\sqrt{/ 1 - x^{2} /}} l e t f i n d λ b y m v c m e t h o d

i f - 1 < x < 1 y = λ {(1 - x^{2})}^{- \frac{1}{2}} s o

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

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

e q u . \Leftrightarrow (1 - x^{2}) λ^{^{'}} {(1 - x^{2})}^{- \frac{1}{2}} + λ x (1 - x^{2}) {(1 - x^{2})}^{- \frac{3}{2}} - λ x {(1 - x^{2})}^{- \frac{1}{2}} = 1

λ^{'} \sqrt{1 - x^{2}} = 1 \Rightarrow λ^{^{'}} = \frac{1}{\sqrt{1 - x^{2}}} \Rightarrow λ = \int \frac{d x}{\sqrt{1 - x^{2}}} + c

λ = a r c s i n x + c a n d y (x) = \frac{1}{\sqrt{1 - x^{2}}} (a r c s i n x + c)

y (x) = \frac{a r c s i n x}{\sqrt{1 - x^{2}}} + \frac{c}{\sqrt{1 - x^{2}}} .

i f 1 - x^{2} < 0 \Leftrightarrow / x / > 1 y = \frac{λ}{\sqrt{x^{2} - 1}} a n d t h e s a m e m e t h o d

g i v e t h e r e s u l t .