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Existence

# Existence - Existence/Uniqueness Recall the theorem 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 . 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 . 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 the theorem is [0 , 1 / 2] , when in fact the solution y ( x ) = tan( x ) exists on the larger interval ( - π/ 2 , π/ 2) .

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

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