Q1-if-y-1-n-y-1-n-2-find-d-2-y-dx-2-

Question Number 7245 by 2402@gmail.com last updated on 18/Aug/16

Q 1 . i f y^{1 / n} + y^{- 1 / n} = 2 f i n d \frac{d^{2} y}{{d x}^{2}} .

Commented by Yozzia last updated on 19/Aug/16

y^{a} + y^{- a} = c (y \neq 0, c \neq 0, a > 0)

I m p l i c i t d i f f e r e n t i a t i o n g i v e s

{a y}^{a - 1} y^{'} - {a y}^{- a - 1} y^{'} = 0

y^{'} ({a y}^{a - 1} - {a y}^{- a - 1}) = 0 (*)

F u r t h e r i m p l i c i t d i f f e r e n t i a t i o n g i v e s

y^{″} ({a y}^{a - 1} - {a y}^{- a - 1}) + y^{'} (a (a - 1) y^{a - 2} y^{'} + a (a + 1) y^{- a - 2} y^{'}) = 0

y^{″} = - \frac{a {(y^{'})}^{2} y^{- 2} ((a - 1) y^{a} + (a + 1) y^{- a})}{{a y}^{- 1} (y^{a} - y^{- a})}

y^{″} = - \frac{{(y^{'})}^{2} y^{- 1} (a (y^{a} + y^{- a}) - y^{a} + y^{- a})}{y^{a} - y^{- a}}

y^{″} = - \frac{{(y^{'})}^{2} y^{- 1} (a c - y^{a} + y^{- a})}{y^{a} - y^{- a}}

I f y^{'} = 0 f r o m (*), \Rightarrow y^{″} = 0 .

I f y^{'} \neq 0 \Rightarrow y^{a} = y^{- a} \Rightarrow y^{″} i s u n d e f i n e d .

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