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 solution of differencial equation y^, =y^2 and y(o)=1.


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

$l e t g i v e t h e s e q u e n c e (y_{n}) / y_{0} (x) = 1 a n d$ $y_{n} (x) = 1 + \int_{0}^{x} {(y_{n - 1} (t))}^{2} d t, l e t s u p p o s e x \in [0, 1] p r o v e$ $t h a t (y_{n}) i s i n c r e a s i n g m a j o r e d b y \frac{1}{1 - x} i f y = {l i m}_{n \to + \infty} y_{n}$ $p r o v e t h a t y i s s o l u t i o n o f d i f f e r e n c i a l e q u a t i o n$ $y^{,} = y^{2} a n d y (o) = 1 .$
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