y-log-x-1-x-2-show-that-x-x-1-2-y-2-x-1-2-y-1-2-

Question Number 52820 by 33 last updated on 13/Jan/19

y = l o g {\sqrt{x} + \frac{1}{\sqrt{x}}}^{2}

s h o w t h a t

x {(x + 1)}^{2} y_{2} + {(x + 1)}^{2} y_{1} = 2

Answered by tanmay.chaudhury50@gmail.com last updated on 13/Jan/19

y = l n (x + 2 + \frac{1}{x}) = l n (\frac{x^{2} + 2 x + 1}{x})

y = 2 l n (x + 1) - l n x

y_{1} = \frac{2}{x + 1} - \frac{1}{x} = \frac{2 x - x - 1}{x^{2} + x} = \frac{x - 1}{x^{2} + x}

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

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

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

{(x + 1)}^{2} y_{1} = (x + 1) (x + 1) \times \frac{x - 1}{x (x + 1)} = \frac{x^{2} - 1}{x} = x - \frac{1}{x}

s o x {(x + 1)}^{2} y_{2} + {(x + 1)}^{2} y_{1}

= - x + 2 + \frac{1}{x} + x - \frac{1}{x}

= 2 h e n c e p r o v e d

Commented by 33 last updated on 14/Jan/19

t h a n k y o u s o m u c h

Commented by tanmay.chaudhury50@gmail.com last updated on 14/Jan/19

m o s t w e l c o m e \dots