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19. x' = -2y, y' = 2x; x(0) = 1, y(0) = 0 %3D

19. x' = -2y, y' = 2x; x(0) = 1, y(0) = 0 %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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## 4.1 Problems In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations. 1. \( x'' + 3x' + 7x = t^2 \) 2. \( x'' + 4x - x^3 = 0 \) (This equation is used in Section 6.4 to describe the motion of a mass connected to a “soft” spring.) 3. \( x'' + 2x' + 26x = 82 \cos 4t \) (This equation was used in Section 3.6 to model the oscillations of a mass-and-spring system.) 4. \( x^{(3)} - 2x'' + x' = 1 + t e^t \) 5. \( x^{(4)} + 3x'' + x = e^t \sin 2t \) 6. \( x^{(4)} + 6x'' - 3x' + x = \cos 3t \) 7. \( t^2 x'' + t x' + (t^2 - 1)x = 0 \)
Transcribed Image Text:## 4.1 Problems In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations. 1. \( x'' + 3x' + 7x = t^2 \) 2. \( x'' + 4x - x^3 = 0 \) (This equation is used in Section 6.4 to describe the motion of a mass connected to a “soft” spring.) 3. \( x'' + 2x' + 26x = 82 \cos 4t \) (This equation was used in Section 3.6 to model the oscillations of a mass-and-spring system.) 4. \( x^{(3)} - 2x'' + x' = 1 + t e^t \) 5. \( x^{(4)} + 3x'' + x = e^t \sin 2t \) 6. \( x^{(4)} + 6x'' - 3x' + x = \cos 3t \) 7. \( t^2 x'' + t x' + (t^2 - 1)x = 0 \)
Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. 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. 17. \( x' = y, \, y' = -x \) 18. \( x' = y, \, y' = x \) 19. \( x' = -2y, \, y' = 2x; \, x(0) = 1, \, y(0) = 0 \) 20. \( x' = 10y, \, y' = -10x; \, x(0) = 3, \, y(0) = 4 \) 21. \( x' = \frac{1}{2} y, \, y' = -8x \) 22. \( x' = 8y, \, y' = -2x \) 23. \( x' = y, \, y' = 6x - y; \, x(0) = 1, \, y(0) = 2 \) 24. \( x' = y, \, y' = 10x - 7y; \, x(0) = 2, \, y(0) = -7 \) 25. \( x' = -y, \, y' = 13x + 4y; \, x(0) = 0, \, y(0) = 3 \) 26. \( x' = y, \, y' = -9x + 6y \)
Transcribed Image Text:Use the method of Examples 6, 7, and 8 to find general solutions of the systems in Problems 17 through 26. 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. 17. \( x' = y, \, y' = -x \) 18. \( x' = y, \, y' = x \) 19. \( x' = -2y, \, y' = 2x; \, x(0) = 1, \, y(0) = 0 \) 20. \( x' = 10y, \, y' = -10x; \, x(0) = 3, \, y(0) = 4 \) 21. \( x' = \frac{1}{2} y, \, y' = -8x \) 22. \( x' = 8y, \, y' = -2x \) 23. \( x' = y, \, y' = 6x - y; \, x(0) = 1, \, y(0) = 2 \) 24. \( x' = y, \, y' = 10x - 7y; \, x(0) = 2, \, y(0) = -7 \) 25. \( x' = -y, \, y' = 13x + 4y; \, x(0) = 0, \, y(0) = 3 \) 26. \( x' = y, \, y' = -9x + 6y \)
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