problemset5

Kn1 m lt n1 lt k l n1 k k0 n t n1 elt m l n1 3

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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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