Solve the ODE (x^2 −2)y′ + xy = 0, with y(1)=1 Mastermind


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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
	$(x^2 −2)y^′ +xy=0 ⇒(x^2 −2)y^′ =−xy ⇒ (y^′ /y)=−(x/(x^2 −2)) ⇒ln∣y∣=−(1/2)ln∣x^2 −2∣ +c ⇒y=(k/( (√(∣x^2 −2∣)))) solution on w={x/x^2 −2<0} ⇒y=(k/( (√(2−x^2 ))))=k(2−x^2 )^(−(1/2)) ⇒y^′ =k^′ (2−x^2 )^(−(1/2)) +k((1/2)2x)(x^2 −2)^(−(3/2)) =k^′ (2−x^2 )^(−(1/2)) +xk(2−x^2 )^(−(3/2)) (e)⇒(x^2 −2)k^′ (2−x^2 )^(−(1/2)) −xk(2−x^2 )^(−(1/2)) +xk(2−x^2 )^(−(1/2)) =0⇒k^′ =0 ⇒k=λ ⇒ y(x)=(λ/( (√(2−x^2 )))) y(1)=1 ⇒λ=1 ⇒y(x)=(1/( (√(2−x^2 ))))$
	$(x^{2} - 2) y^{^{'}} + x y = 0 \Rightarrow (x^{2} - 2) y^{^{'}} = - x y \Rightarrow$ $\frac{y^{^{'}}}{y} = - \frac{x}{x^{2} - 2} \Rightarrow l n ∣ y ∣= - \frac{1}{2} l n ∣ x^{2} - 2 ∣ + c$ $\Rightarrow y = \frac{k}{\sqrt{∣ x^{2} - 2 ∣}}$ $s o l u t i o n o n w = {x / x^{2} - 2 < 0}$ $\Rightarrow y = \frac{k}{\sqrt{2 - x^{2}}} = k {(2 - x^{2})}^{- \frac{1}{2}}$ $\Rightarrow y^{^{'}} = k^{^{'}} {(2 - x^{2})}^{- \frac{1}{2}} + k (\frac{1}{2} 2 x) {(x^{2} - 2)}^{- \frac{3}{2}}$ $= k^{^{'}} {(2 - x^{2})}^{- \frac{1}{2}} + x k {(2 - x^{2})}^{- \frac{3}{2}}$ $(e) \Rightarrow (x^{2} - 2) k^{^{'}} {(2 - x^{2})}^{- \frac{1}{2}}$ $- x k {(2 - x^{2})}^{- \frac{1}{2}} + x k {(2 - x^{2})}^{- \frac{1}{2}}$ $= 0 \Rightarrow k^{^{'}} = 0 \Rightarrow k = λ \Rightarrow$ $y (x) = \frac{λ}{\sqrt{2 - x^{2}}}$ $y (1) = 1 \Rightarrow λ = 1 \Rightarrow y (x) = \frac{1}{\sqrt{2 - x^{2}}}$
	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

	$n o$
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