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Chapter 7 Linear Systems of Differential Equations Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. 11. x' = y, y' = -x 12. x' = y, y' = x 13. x' = -2y, y' = 2x; x(0) = 1, y(0) = 0 14. x' = 10y, y' = -10x; x(0) = 3, y(0) = 4 15. x' = 1/2y, y' = -8x 16. x' = 8y, y' = -2x 17. x' = y, y' = 6x - y; x(0) = 1, y(0) = 2 18. x' = -y, y' = 10x - 7y; x(0) = 2, y(0) = -7 19. x' = -y, y' = 13x + 4y; x(0) = 0, y(0) = 3 20. x' = y, y' = -9x + 6y 21. (a) Calculate [x(t)]^2 + [y(t)]^2 to show that the trajectories of the system x' = y, y' = -x of Problem 11 are circles. (b) Calculate [x(t)]^2 - [y(t)]^2 to show that the trajectories of the system x' = y, y' = x of Problem 12 are hyperbolas. 22. (a) Beginning with the general solution of the system x' = -2y, y' = 2x of Problem 13, calculate x^2 + y^2 to show that the trajectories are circles. (b) Show similarly that the trajectories of the system x' = 1/2y, y' = -8x of Problem 15 are ellipses with equations of the form

          Chapter 7 Linear Systems of Differential Equations
Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
11. x' = y, y' = -x
12. x' = y, y' = x
13. x' = -2y, y' = 2x; x(0) = 1, y(0) = 0
14. x' = 10y, y' = -10x; x(0) = 3, y(0) = 4
15. x' = 1/2y, y' = -8x
16. x' = 8y, y' = -2x
17. x' = y, y' = 6x - y; x(0) = 1, y(0) = 2
18. x' = -y, y' = 10x - 7y; x(0) = 2, y(0) = -7
19. x' = -y, y' = 13x + 4y; x(0) = 0, y(0) = 3
20. x' = y, y' = -9x + 6y
21. (a) Calculate [x(t)]^2 + [y(t)]^2 to show that the trajectories of the system x' = y, y' = -x of Problem 11 are circles. (b) Calculate [x(t)]^2 - [y(t)]^2 to show that the trajectories of the system x' = y, y' = x of Problem 12 are hyperbolas.
22. (a) Beginning with the general solution of the system x' = -2y, y' = 2x of Problem 13, calculate x^2 + y^2 to show that the trajectories are circles. (b) Show similarly that the trajectories of the system x' = 1/2y, y' = -8x of Problem 15 are ellipses with equations of the form

Chapter 7 Linear Systems of Differential Equations
Use the method of Examples 5, 6, and 7 to find general solutions of the systems in Problems 11 through 20. If initial conditions are given, find the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.
11. x' = y, y' = -x
12. x' = y, y' = x
13. x' = -2y, y' = 2x; x(0) = 1, y(0) = 0
14. x' = 10y, y' = -10x; x(0) = 3, y(0) = 4
15. x' = 1/2y, y' = -8x
16. x' = 8y, y' = -2x
17. x' = y, y' = 6x - y; x(0) = 1, y(0) = 2
18. x' = -y, y' = 10x - 7y; x(0) = 2, y(0) = -7
19. x' = -y, y' = 13x + 4y; x(0) = 0, y(0) = 3
20. x' = y, y' = -9x + 6y
21. (a) Calculate [x(t)]^2 + [y(t)]^2 to show that the trajectories of the system x' = y, y' = -x of Problem 11 are circles. (b) Calculate [x(t)]^2 - [y(t)]^2 to show that the trajectories of the system x' = y, y' = x of Problem 12 are hyperbolas.
22. (a) Beginning with the general solution of the system x' = -2y, y' = 2x of Problem 13, calculate x^2 + y^2 to show that the trajectories are circles. (b) Show similarly that the trajectories of the system x' = 1/2y, y' = -8x of Problem 15 are ellipses with equations of the form

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