if-y-sin-1-x-2-check-whether-1-x-2-y-n-2-2n-1-xy-n-1-n-2-y-n-0-or-not-

Question Number 25569 by rita1608 last updated on 11/Dec/17

i f y = {({s i n}^{- 1} x)}^{2}, c h e c k w h e t h e r

(1 - x^{2}) y_{n + 2} - (2 n + 1) {x y}_{n + 1} + n^{2} y_{n} = 0

o r n o t .

Commented by prakash jain last updated on 12/Dec/17

y e s . The ans is no .

Commented by rita1608 last updated on 11/Dec/17

i n t h i s q u e s t i o n i a m g e t t i n g a n s

(1 - x^{2}) y_{n + 2} (2 n - 1) {x y}_{n + 1} - n^{2} y_{n} = 0

i j u s t w a n t t o k n o w w h y i a m g e t t i n g

- v e s i g n b e f o r e n^{2} y_{n}

Commented by prakash jain last updated on 11/Dec/17

Can u please share your workings ?

Commented by rita1608 last updated on 12/Dec/17

o k

Commented by prakash jain last updated on 12/Dec/17

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

y^{'} = \frac{{2 \sin}^{- 1} x}{\sqrt{1 - x^{2}}}

y^{″} = \frac{2}{(1 - x^{2})} + \frac{2 x \sin^{- 1} x}{{(1 - x^{2})}^{3 / 2}}

y_{3} = \frac{6 x}{{(1 - x^{2})}^{2}} + \frac{6 x^{2} \sin^{- 1} x}{{(1 - x^{2})}^{5 / 2}} + \frac{{2 \sin}^{- 1} x}{{(1 - x^{2})}^{3 / 2}}

(1 - x^{2}) y_{3} - (2 n + 1) {x y}_{2} - n^{2} y_{1}

= \frac{6 x}{(1 - x^{2})} + \frac{6 x^{2} \sin^{- 1} x}{{(1 - x^{2})}^{3 / 2}} + \frac{{2 \sin}^{- 1} x}{{(1 - x^{2})}^{1 / 2}}

- \frac{6 x}{(1 - x^{2})} - \frac{6 x^{2} \sin^{- 1} x}{{(1 - x^{2})}^{3 / 2}}

- \frac{{2 \sin}^{- 1} x}{\sqrt{1 - x^{2}}} = 0

I am also getting - n^{2} for n = 1

Commented by rita1608 last updated on 12/Dec/17

Commented by prakash jain last updated on 12/Dec/17

It looks correct . Also i checked

calculation with y_{3}, y_{2}, y_{1} it is

- n^{2} .

Commented by rita1608 last updated on 12/Dec/17

o k b u t i n q u e s t i o n t h e y a r e s a y i n g t o

c h e c k . t h e n a n s i s n o ?