nagle_differential_equations_ISM_Part38

nagle_differential_equations_ISM_Part38 - Exercises 5.4 for...

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Unformatted text preview: Exercises 5.4 for y < 0, | x | > 1, y =- c x 2- 1; for y < 0, | x | < 1, y =- c 1- x 2 ; all with c If c = 1, y = 1- x 2 are semicircles ending at (1 , 0) and (- 1 , 0) 12. 9 x 2 + 4 y 2 = c . See Fig. 5B on page 187 14. y = cx 2 / 3 See Fig. 5C on page 187 16. (0 , 0) is a stable node. See Fig. 5D on page 188 18. (0 , 0) is an unstable node; (0 , 5) is a stable node; (7 , 0) is a stable node; (3 , 2) is a saddle point; See Fig. 5E on page 188 20. y = v v =- y (0 , 0) is a center. See Fig. 5F on page 189 22. y = v v =- y 3 (0 , 0) is a center. See Fig. 5G on page 189 24. y = v v =- y + y 3 (0 , 0) is a center; (- 1 , 0) is a saddle point; (1 , 0) is a saddle point. See Fig. 5H on page 190 26. x 2 2 + x 4 4 + y 2 2 = c ; all solutions are bounded. See Fig. 5I on page 190 28. (0 , 0) is a center; (1 , 0) is a saddle point 30. (a) x x * , y y * , f and g are continuous implies x ( t ) f ( x ( t ) ,y ( t )) f ( x * ,y * ) and y ( t ) g ( x ( t ) ,y ( t )) g ( x , y * ) (b) x ( t ) = t Z T x ( ) d + t ( T ) > f ( x * ,y * ) 2 ( t- T ) + x ( T ) f ( x * ,y * ) t 2 + C 181 Chapter 5 (c) If f ( x * ,y * ) > 0, f ( x * ,y * ) t implying x ( t ) (d) Similar (e)...
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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_Part38 - Exercises 5.4 for...

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