[tex]y(t) = y0 +(adsbygoogle = window.adsbygoogle || []).push({});

\int^{ t}_{t_0}

f(s, y(s)) ds.[/tex]

Picard’s method starts with the deﬁnition of what it means to be a solution: if you guess that a function

φ(t) is a solution, then you can check your guess by substituting it into the right-hand side of equation (2) and

comparing it to the left-hand side, which is simply φ(t) itself. The new idea is that the process of checking

each guess produces a new guess which, even if it is not the correct solution, is a better approximation

than the one you started with. In this way we obtain an iterative solution, with each new approximation

computed from the previous one by the right-hand side of equation (2). This should be reminiscent of

Newton’s method. In fact, the proof that Picard’s method produces a convergent sequence is similar to the

proof for Newton’s method.

How can I be sure that nth solution is better than n-1th solution. Is there some easy way to see this? Is there a case where this is incorrect?

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# Picard method of succesive approximation

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