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Unformatted text preview: + nh, y n ) h t n = t + nh This is known as Eulers Method . The smaller the step size h , the better the approximation. 2.8: The Existence and Uniqueness Theorem. Consider the initial value problem dy dt = G ( t,y ) , y ( t ) = y . In order to prove the Existence and Uniqueness Theorems of 2.4, one constructs a sequence of approximate solutions ( t ) , 1 ( t ) , ..., n ( t ) , ... such that the limit ( t ) = lim n n ( t ) is the actual solution. To this end, dene the constant function ( t ) = y ; and dene recursively the sequence of functions n +1 ( t ) = t t G , n ( ) d + y . This is known as the Method of Successive Approximations or the Method of Picard Iterations . The functions n ( t ) are good approximations to the actual solution ( t )....
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.
- Spring '09