let-give-the-sequence-y-n-y-0-x-1-and-y-n-x-1-0-x-y-n-1-t-2-dt-let-suppose-x-0-1-prove-that-y-n-is-increasing-majored-by-1-1-x-if-y-lim-n-y-n-prove-that-y-is-solu

Question Number 29037 by abdo imad last updated on 03/Feb/18

l e t g i v e t h e s e q u e n c e (y_{n}) / y_{0} (x) = 1 a n d

y_{n} (x) = 1 + \int_{0}^{x} {(y_{n - 1} (t))}^{2} d t, l e t s u p p o s e x \in [0, 1] p r o v e

t h a t (y_{n}) i s i n c r e a s i n g m a j o r e d b y \frac{1}{1 - x} i f y = {l i m}_{n \to + \infty} y_{n}

p r o v e t h a t y i s s o l u t i o n o f d i f f e r e n c i a l e q u a t i o n

y^{,} = y^{2} a n d y (o) = 1 .