Chapter9diffeq

Chapter9diffeq - 825 9.1 Iterative Equations First-Order...

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Unformatted text preview: 825 9.1 Iterative Equations First-Order Iterative Equations For Problems 1-12 we use the fact that the solution of 1 n n y a y b + = + is 1 . 1 n n n a y a y b a = + 1. 1 1 2 3 n n y y + = + 1 y = (b) 6 n 10 y n (a),(c) Because 1 2 , 3 , 1 , a b y = = = 1 2 1 2 1 1 3 2 1 5 6 2 n n n n y = + = + As n increases, the orbit or solution approaches 6 from below. Thus 6 . n e y y = 1 1 2 3, 1 n n y y y + = + = 2. 1 1 2 3 n n y y + = + 1 y = (b) n 10 y n 1 7 (a),(c) Because 1 2 , 3 , 1 , a b y = = = 1 2 1 2 1 1 3 2 1 7 6 2 n n n n y = + = + As n increases, the orbit or solution approaches 6 from below. Thus 6 . n e y y = 1 1 2 3, 1 n n y y y + = + = CHAPTER 9 CHAPTER Discrete Dynamical Systems 826 CHAPTER 9 Discrete Dynamical Systems 3. 1 1 2 3 n n y y + = + 1 y = (b) 3 n 10 y n (a),(c) Because 1 , 3 , 1 , 2 a b y = = = ( ) 1 2 3 2 1 1 3 2 1 2 2 n n n n y = + = + As n increases, the orbit approaches 2, with successive n y alternatively above and below this value. Thus 2 . n e y y = 1 1 2 3, 1 n n y y y + = + = 4. 1 1 2 3 n n y y + = + 1 y = (b) 4 n 10 y n (a),(c) Because 1 , 3 , 1 , 2 a b y = = = 1 1 1 2 3 3 2 2 1 3 2 2 n n n n y = + = + As n increases, the orbit approaches 2, with successive n y alternatively above and below this value. Thus 2 . n e y y = 1 1 2 3, 1 n n y y y + = + = 5. 1 2 3 n n y y + = + 1 y = (b) 5000 n 10 y n (a),(c) Because 2, 3, 1, a b y = = = 2 1 2 3 1 4 2 3 n n n n y = + = As n increases, the solution grows without bound. 1 2 3 , 1 n n y y y + = + = SECTION 9.1 Iterative Equations 827 6. 1 2 3 n n y y + = + 1 y = (b) 2500 y n n 10 (a),(c) Because 2, 3, 1, a b y = = = 2 1 2 3 1 2 3 n n n n y = + = As n increases, the solution grows without bound. 1 2 3 , 1 n n y y y + = + = 7. 1 2 3 n n y y + = + 1 y = (b) 1.25 n 10 y n (a),(c) Because 2, 3, 1, a b y = = = ( 2) 1 ( 2) 3 3 1 for all n n n y n = + = The orbit starts at y and remains there, because this is a fixed point. It is, however, a repelling fixed point, so at any other value for y the solution is unbounded. See Problem 8....
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Chapter9diffeq - 825 9.1 Iterative Equations First-Order...

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