Department of Mathematical Sciences University of Delaware Prof. T. Angell September 23, 2009 Mathematics 351 Exercise Sheet 3 Exercise 11 : A growth model that is used in such diverse Felds as Economics and Medicine (in this latter case, to model the size of animal tumors) is the equation dx dt = rx ln ° K x ± , where r and K are positive constants. (a) Using an argument similar to that for the Logistic Equation, show that any population satisfying x (0) = x o > 0 will satisfy lim t →∞ x ( t )= K . (b) Letting y =ln( x ), Fnd the exact solution satisfying x (0) = x o >K . Exercise 12 : (Another part of the Principle of Superposition .) Suppose that x 1 ( t ) is a solution of the di±erential equation ˙ x = a ( t ) x + b 1 ( t ) and that x 2 ( t ) is a solution of ˙ x = a ( t ) x ( t )+ b 2 ( t ). Show that the sum of the two functions x 1 ( t )+ x 2 ( t ) is a solution of the di±erential equation ˙ x = a ( t ) x +[ b 1 ( t )+ b 2 ( t )]. Exercise 13: The secretion of hormones into the blood is often a periodic activity. If a hormone
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