Unformatted text preview: �
x(t) − xn (t) = � (xk (t) − xk−1 (t))�
� =n+1
� k
∞
�
≤
xk (t) − xk−1 (t)
≤
=
= k=n+1
∞
M � (LT )k
L
k!
k=n+1
∞
M (LT )n+1 � (LT )k
L (n+1)!
k!
k=0
n T n+1 eLT
M L (n+1)! 3. (a) (⇒) Let F (t) = f (t, φ(t)) and solve x�� = F (t) when x(t0 ) = x0 ,
x� (t0 ) = x1 .
� � t
d t
∂f (⇐ Use )
f (s, t) ds = f (t, t) + (s, t) ds. dt t0
t0 ∂t (b) Repeat the proof of the local existence theorem by showing
�� t
�
�
�
� (t − s)f (s, xn−1 (s) ds�
(1) xn (t) − x0  = x1  t − t0  + �
� t...
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This note was uploaded on 01/22/2014 for the course MATH 18.034 taught by Professor Hur during the Spring '07 term at MIT.
 Spring '07
 HUR
 Differential Equations, Equations

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