Solve-the-ODE-x-2-2-y-xy-0-with-y-1-1-Mastermind-

Question Number 169051 by Mastermind last updated on 23/Apr/22

S o l v e t h e O D E

(x^{2} - 2) y^{'} + x y = 0, w i t h y (1) = 1

M a s t e r m i n d

Commented by haladu last updated on 23/Apr/22

y^{'} (x^{2} - 2) = - xy

\frac{dy}{y} = - \frac{x dx}{x^{2} - 2}

\int \frac{dy}{y} = - \frac{1}{2} \int \frac{2 x}{x^{2} - 2}

\ln (y) = - \frac{1}{2} \ln (x^{2} - 2) + C

y = x = 1

\ln (1) = - \frac{1}{2} \ln (1^{2} - 2) + C

0 = - \frac{1}{2} \ln (- 1) + C

C = \frac{1}{2} \ln (- 1)

\ln (y) = - \frac{1}{2} \ln (x^{2} - 2) + \frac{1}{2} \ln (- 1)

- 2 \ln (y) = \ln (x^{2} - 2) - \ln (- 1)

\frac{1}{y^{2}} = \frac{x^{2} - 2}{- 1}

y^{2} = 2 - x^{2}

Commented by Mastermind last updated on 23/Apr/22

C o u l d y o u p l e a s e r e c h e c k y o u r s o l u t i o n ?

Commented by haladu last updated on 23/Apr/22

y e s i was in hurry becuse it was time for sallat .

Commented by Mastermind last updated on 24/Apr/22

O k a y i u n d e r s t a n d

Answered by Mathspace last updated on 23/Apr/22

(x^{2} - 2) y^{^{'}} + x y = 0 \Rightarrow (x^{2} - 2) y^{^{'}} = - x y \Rightarrow

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

\Rightarrow y = \frac{k}{\sqrt{∣ x^{2} - 2 ∣}}

s o l u t i o n o n w = {x / x^{2} - 2 < 0}

\Rightarrow y = \frac{k}{\sqrt{2 - x^{2}}} = k {(2 - x^{2})}^{- \frac{1}{2}}

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

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

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

- x k {(2 - x^{2})}^{- \frac{1}{2}} + x k {(2 - x^{2})}^{- \frac{1}{2}}

= 0 \Rightarrow k^{^{'}} = 0 \Rightarrow k = λ \Rightarrow

y (x) = \frac{λ}{\sqrt{2 - x^{2}}}

y (1) = 1 \Rightarrow λ = 1 \Rightarrow y (x) = \frac{1}{\sqrt{2 - x^{2}}}

Commented by Mastermind last updated on 24/Apr/22

S i r, i t h i n k a n s w e r i s y = \frac{i}{\sqrt{x^{2} - 2}}

Commented by Mathspace last updated on 24/Apr/22

n o