examples152-ch1

examples152-ch1 - MATH 152 Spring 2004-05 Applied Linear...

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Unformatted text preview: MATH 152 Spring 2004-05 Applied Linear Algebra & Differential Equations Worked Examples Dr. Tony Yee Department of Mathematics The Hong Kong University of Science and Technology January 29, 2005 ii Contents Table of Contents iii 1 Introduction 1 iii Chapter 1 Introduction ¥ Example 1.1 (Linear vs. nonlinear differential equations) Indicate whether the following equation is linear or nonlinear. 1. y + y + 1 = 0. 2. y + y = ye t . 3. ( y ) 2 + y = t + 1. 4. 7 y + t- 1 y = 1 , t > 0. 5. dy dt + t = e 3 y . 6. y + 2 y = e- t . 7. y dy dt + ty = 1. 8. y + ty = (1 + y 2 ). 9. y + 4 y = sin- 1 t , | t | < 1. 10. ( t 2 + 1) dy dt + 2 y = 0. 11. y + 2 y = e t . 12. y + 2 t 2 y = t . 13. ( y ) 3 + y = t . 14. y + e t (1 + y 00 ) = 0. 15. y + ty = (1 + √ y ). 16. dy dt + 2 y 3 = 0. 17. dy dt + t = y π . 18. y + ry = k 3 , r,k constants . 19. dy dt + πy = 0. 20. (1 + t 3 ) y + y = 0. 21. y + √ 1- t 2 y = t , | t | < 1. 22. ty + 2 y = te t , t 6 = 0. 23. dy dt = ln( t + y ). 24. dy dt = e t + y . Solution 1. linear 2. linear 3. nonlinear 4. linear 5. nonlinear 6. nonlinear 7. nonlinear 8. nonlinear 9. linear 10. linear 11. linear 12. linear 13. nonlinear 14. linear 15. nonlinear 16. nonlinear 17. nonlinear 18. linear 19. linear 20. linear 21. linear 22. linear 23. nonlinear 24. nonlinear 2 1 1. Introduction ¥ Example 1.2 (Solutions of ODEs) Show that y ( t ) = t- 3 / 2 is a solution to 4 t 2 y 00 + 12 ty + 3 y = 0 for t >...
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examples152-ch1 - MATH 152 Spring 2004-05 Applied Linear...

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