E x for all x thus the picard iterates converge

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= e x for all x . Thus the Picard iterates converge everywhere to a function e x which we can easily see is the solution to the IVP. In general we cannot expect the Picard iterates to converge for all x , but the following theorem gives us conditions under which the Picard iterates are guaranteed to converge. 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 ] . This important theorem will be covered in more depth in the next lecture.
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