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Unformatted text preview: Picard Iterates From our study of exact equations, it becomes apparent that very few differential equations (hence IVP’s) can be solved explicitly. Soon, we will study methods for numerically approximating solutions to an IVP, and in any practical application these will serve us just as well as the actual solution. But before we can begin attempts to numerically approximate a solution, we have to make sure that a unique solution actually exists. We will consider the IVP dy dx = f ( x,y ) , y ( x ) = y . Outline for proving the existence of a solution: I. Find a sequence of functions y ( x ) ,y 1 ( x ) ,y 2 ( x ) ,... that are better and better approximations to a solution of the IVP. II. Show that on some interval [ x ,x + α ] , this sequence of functions has a limit y ( x ) = lim n →∞ y n ( x ) . III. Show that y ( x ) is a solution to the IVP on this interval. Remark: We will not be able to explicitly find the limiting function y ( x ) , but it is enough that we can prove it exists and that it satisfies the IVP. This justifies our attempt to use numerical methodscan prove it exists and that it satisfies the IVP....
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