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### nagle_differential_equations_ISM_Part65

Course: MA 221, Spring 2011
School: Stevens Institute of...
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Word Count: 165

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12 4 2 K4 K2 0 2 4 K2 Chapter K4 Figure 12S: Phase plane diagram in Problem 6, Review Section 4 y 2 K2 K1 0 1 2 x 3 2 x K2 Figure 12T: Potential and Phase plane diagrams in Problem 8, Review Section 4 2 K4 K2 0 2 4 x K2 K4 Figure 12U: Phase plane diagram in Problem 14, Review Section 316 CHAPTER 13: Existence and Uniqueness Theory EXERCISES 13.1: x Introduction: Successive...

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12 4 2 K4 K2 0 2 4 K2 Chapter K4 Figure 12S: Phase plane diagram in Problem 6, Review Section 4 y 2 K2 K1 0 1 2 x 3 2 x K2 Figure 12T: Potential and Phase plane diagrams in Problem 8, Review Section 4 2 K4 K2 0 2 4 x K2 K4 Figure 12U: Phase plane diagram in Problem 14, Review Section 316 CHAPTER 13: Existence and Uniqueness Theory EXERCISES 13.1: x Introduction: Successive Approximations 2. y (x) = sin[t + y (t)] dt x 4. y (x) = 1 + 0 ey(t) dt 6. 0.3775396 8. 2.2360680 10. 1.9345632 12. y1 (x) = 1 + x; y2 (x) = 1 + x + x2 + 14. y1 (x) = y2 (x) = sin x 16. y1 (x) = y2 (x) = 3 2 5 3 x+ 2 3 1 2 x2 ; 1 6 x3 1 3 x3 x + x2 EXERCISES 13.2: 2. No 4. Yes 6. Yes Picards Existence and Uniqueness Theorem 14. No. Let yn (x) = Then n n2 x , 0 x (1/n) 2n n2 x , (1/n) x (2/n) 0 , 2/n x 1 . lim yn (x) = 0 , 317 Chapter 13 but 1 n 0 lim yn (x) dx = 1 = 0 . EXERCISES 13.3: 2. [2, 1) 4. (0, 3] 6. (, ) EXERCISES 13.4: 2. 102 e 4. 102 e 6. 102 e 8. 10. e 6 1 24 esin 1 2 e1/2 Existence of Solutions of Linear Equations Continuous Dependence of Solutions REVIEW PROBLEMS 2. 0.7390851 x 4. 9 + 0 [t2 y 3 (t) y 2 (t)] dt 6. y1 (x) = 1 + 2x; y2 (x) = 1 + 2x 2x2 8. No 10. 12. e 6 , 22 318
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