solve: y′′(1 + 4x^2 ) − 8y = 0


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Question Number 38802 by tawa tawa last updated on 30/Jun/18

$solve : y^{″} (1 + {4 x}^{2}) - 8 y = 0$
	Answered by MrW3 last updated on 30/Jun/18

	$y = {a x}^{2} + b x + c$ $y^{'} = 2 a x + b$ $y^{″} = 2 a$ $2 a (1 + 4 x^{2}) - 8 ({a x}^{2} + b x + c) = 0$ $8 {a x}^{2} + 2 a - 8 {a x}^{2} - 8 b x - 8 c = 0$ $- 8 b x + 2 a - 8 c = 0$ $\Rightarrow - 8 b = 0 \Rightarrow b = 0$ $\Rightarrow 2 a - 8 c = 0 \Rightarrow a = 4 c$ $\Rightarrow y = c (4 x^{2} + 1)$
	Commented by tawa tawa last updated on 30/Jun/18

	$God bless you sir .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 30/Jun/18

	$y = x^{α} + h b y t r i a l m e t h o d$ $y = x^{2} + \frac{1}{4} i s a s o l u t i o n . . .$ $\frac{d y}{d x} = 2 x \frac{d^{2} y}{{d x}^{2}} = 2$ $2 (1 + 4 x^{2}) - 8 (x^{2} + \frac{1}{4}) = 0$ $S O y = x^{2} + \frac{1}{4} i s t h e s o l u t i o n$ $p l s g i v e m e t i m e t o s o l v e b y s t a n d a r d m e t h o d$
	Commented by tawa tawa last updated on 30/Jun/18

	$i will be waiting sir . God bless you$
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