Existence - Existence/Uniqueness Recall the theorem:...

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Existence/Uniqueness Recall the theorem: Theorem Given the IVP y 0 = f ( x,y ) , y ( x 0 ) = y 0 , suppose f and ∂f/∂y are continuous in the rectangle R given by x 0 x x 0 + a | y - y 0 | ≤ b. Compute the values M = max ( x,y ) in R | f ( x,y ) | , α = min ± a, b M ² . Then the Picard Iterates converge to a unique solution on the interval [ x 0 ,x 0 + α ] . The next example shows how that the interval predicted by the theorem will be smaller than the actual interval of existence of the solution: I Example What is the largest interval of existence predicted by the above theorem for the solution to the IVP y 0 = 1 + y 2 , y (0) = 0? Solution Consider the rectangle R : 0 x a | y | ≤ b. We compute M = max ( x,y ) in R | f ( x,y ) | = max ( x,y ) in R 1 + y 2 = 1 + b 2 and then α = min ± a, b 1 + b 2 ² . Since 1 / (1+ b 2 ) has a maximum value of 1 / 2 , α can be at most 1 / 2 (and it attains this value, for example pick a = 1 and b = 1 ). Then the largest interval of existence guaranteed by
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This note was uploaded on 03/03/2012 for the course MATH 4430 at Colorado.

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Existence - Existence/Uniqueness Recall the theorem:...

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