Unformatted text preview: to compute y n ( x ) for n = 1 , 2 , 3 for the initial value problem y ( x ) = x + y ( x ) , y (0) = 1 . Compute the exact solution and compare y 1 , y 2 and y 3 to the appropriate partial sums of the Taylor series for the exact solution. 4. ( 5 marks ) Consider the initial value problem y 000 + y 2 = 1 ,y (1) = 1 ,y (1) = 0 ,y 00 (1) = 0 a) Rewrite this equation in an equivalent vector initial value problem Y = F ( Y ( x ) ,x ) , Y ( x ) = Y b) Use the deﬁnition to show that F is Lipschitz continuous with respect to Y in any region [ a,b ] × R 2 × R ⊂ R 3 × R , where a,b are ﬁnite real numbers. c) Apply the appropriate theorem to predict where a solution of the initial value problem exists. 1...
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- Spring '08