nagle_differential_equations_ISM_Part65

# nagle_differential_equations_ISM_Part65 - lim n →∞ y n...

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Chapter 12 K 4 K 2 0 2 4 K 4 K 2 2 4 Figure 12–S : Phase plane diagram in Problem 6, Review Section x K 2 K 1 0 1 2 3 y 2 4 x K 2 2 Figure 12–T : Potential and Phase plane diagrams in Problem 8, Review Section x K 4 K 2 0 2 4 K 4 K 2 2 4 Figure 12–U : Phase plane diagram in Problem 14, Review Section 316

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CHAPTER 13: Existence and Uniqueness Theory EXERCISES 13.1: Introduction: Successive Approximations 2. y ( x ) = x R π sin[ t + y ( t )] dt 4. y ( x ) = 1 + x R 0 e y ( t ) dt 6. 0 . 3775396 8. 2 . 2360680 10. 1 . 9345632 12. y 1 ( x ) = 1 + x ; y 2 ( x ) = 1 + x + x 2 + ± 1 3 ² x 3 14. y 1 ( x ) = y 2 ( x ) = sin x 16. y 1 ( x ) = ± 3 2 ² - x + ± 1 2 ² x 2 ; y 2 ( x ) = ± 5 3 ² - ± 3 2 ² x + x 2 - ± 1 6 ² x 3 EXERCISES 13.2: Picard’s Existence and Uniqueness Theorem 2. No 4. Yes 6. Yes 14. No. Let y n ( x ) = n 2 x, 0 x (1 /n ) 2 n - n 2 x, (1 /n ) x (2 /n ) 0 , 2 /n x 1 . Then
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Unformatted text preview: lim n →∞ y n ( x ) = 0 , 317 Chapter 13 but lim n →∞ 1 Z y n ( x ) dx = 1 6 = 0 . EXERCISES 13.3: Existence of Solutions of Linear Equations 2. [-2 , 1) 4. (0 , 3] 6. (-∞ , ∞ ) EXERCISES 13.4: Continuous Dependence of Solutions 2. 10-2 e 4. 10-2 e √ 2 e-1 / 2 6. 10-2 e 8. ± 1 24 ² e sin 1 10. e 6 REVIEW PROBLEMS 2. . 7390851 4. 9 + x Z [ t 2 y 3 ( t )-y 2 ( t )] dt 6. y 1 ( x ) =-1 + 2 x ; y 2 ( x ) =-1 + 2 x-2 x 2 8. No 10. ³-π 2 , π 2 ´ 12. e 6 318...
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## This note was uploaded on 02/12/2011 for the course MA 221 taught by Professor Mazmani during the Spring '08 term at Stevens.

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nagle_differential_equations_ISM_Part65 - lim n →∞ y n...

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