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hw3(4) - Prove that the polygonal region R depicted below...

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Homework assignment 3 * Due date: Friday November 9, 2007 1. Consider the following model of a bioreactor with two species with concentrations x and y : ˙ x = x ( f (1 - x - y ) - 1) ˙ y = y ( g (1 - x - y ) - 1) , where ( x, y ) T = { ( x, y ) R 2 | x > 0 , y > 0 , x + y < 1 } . The functions f and g are differentiable functions with f (0) = g (0) = 0 with f , g > 0. Show that this system has no non-trivial periodic solutions in T . 2. In class we discussed the van der Pol oscillator which we showed could be written as follows: ˙ x = x + y - x 3 / 3 ˙ y = - x We showed that the van der Pol oscillator has a nontrivial periodic solution in a certain annular region that did not contain the origin. We did not completely prove that this region is a trapping region, and the purpose of this problem is to fill the gap by doing the following:
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Unformatted text preview: Prove that the polygonal region R depicted below is a trapping region. 3. Find the ﬁrst three terms of two linearly independent solutions of 2 x ( x-1) y 00 + 3( x-1) y-y = 0 , x > . 4. Consider Laguerre’s equation: xy 00 + (1-x ) y + ny = 0 , where n is a nonnegative integer. Show that for every n , this equation has a polynomial solution L n ( x ) and determine L , L 1 , L 2 and L 3 (This is problem 24 from section 8.7 in our text). 5. Do problem 22 of section 8.7 in our text. * MAP 4305; Instructor: Patrick De Leenheer. R − 3 3 y x − 6 6 − 3 3 1...
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