The functions that l leaves unchanged provided y x is

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the functions that L leaves unchanged (provided y ( x ) is continuous). In other words, we can rewrite equation (1) as y ( x ) = L [ y ( x )] . This motivates the following definition: Definition Given the IVP mentioned above, we can define a sequence of functions recursively by setting y 0 ( x ) = y 0 and for n 0 defining y n +1 ( x ) = L [ y n ( x )] . The functions y n ( x ) for n 0 are called Picard iterates . Remark: It may seem slightly confusing that y 0 is being used to denote both the first Picard iterate (which is a function) and the constant y 0 from the initial condition (which is a number). But it should be clear from the context whether we mean the constant function y 0 ( x ) or the number y 0 .
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I Example Compute the first three Picard iterates for the IVP: y 0 = y, y (0) = 1 . Solution The first Picard iterate is by definition the constant function y 0 ( x ) = 1 . To find the next two, we need to compute the operator L to be L [ y ( x )] = y 0 + Z x x 0 f ( t, y ( t )) dt = 1 + Z x 0 y ( t ) dt Now we compute y 1 ( x ) = L [ y 0 ( x )] = 1 + Z x 0 1 dt = 1 + x and also y 2 ( x ) = 1 + Z x 0 1 + t dt = 1 + x + x 2 / 2 In fact, in the previous example it is not hard to see that lim n →∞ y n ( x ) = X k =0 x k k !
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