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Assignment 2

# Assignment 2 - to compute y n x for n = 1 2 3 for the...

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AM351 Spring 08 Due Friday, June 20 Assignment 2 Instructions: write your solutions clearly, explain your steps carefully and refer to the- orems and lemmas when needed. The assignment is due either in class or in the as- signments drop box (4th ﬂoor) by 2pm. 1. ( 5 marks ) Consider the system of diﬀerential equations with constant coeﬃcients Y 0 = AY + f ( x ) or y 0 1 = a 11 y 1 + a 12 y 2 + f 1 ( x ) y 0 2 = a 21 y 1 + a 22 y 2 + f 2 ( x ) Assuming that a ij R , y 1 and y 2 are two times diﬀerentiable and f 1 , f 2 are dif- ferentiable, show that this equation may be reduced to the second order equation y 00 1 + α 1 y 0 1 + α 2 y 1 = g ( x ) , where α 1 = Trace( A ), α 2 = det( A ) and g ( x ) depends on f 1 and f 2 . Hint: Diﬀerentiate the ﬁrst equation with respect to x, then use the ﬁrst and second equation to eliminate y 2 and y 0 2 from this equation. 2. ( 5 marks ) Solve the diﬀerential equation Y 0 = AY + F ( x ) where A = ± 3 -1 1 1 ² , F ( x ) = ± e x e 2 x ² 3. ( 5 marks ) Apply the method of successive approximations (Picard iterations)
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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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