odes - Introduction to differential equations and...

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Unformatted text preview: Introduction to differential equations and applications Bernard Deconinck Department of Applied Mathematics University of Washington Campus Box 352420 Seattle, WA, 98195, USA August 13, 2006 Prolegomenon These are the lecture notes for Amath 351: Introduction to differential equations and applications. This is the first year these notes are typed up, thus it is guaranteed that these notes are full of mistakes of all kinds, both innocent and unforgivable. Please point out these mistakes to me so they may be corrected for the benefit of your successors. If you think that a different phrasing of something would result in better understanding, please let me know. The figures in these lectures were produced using John Polkings DFIELD2005.10 and PPLANE2005.10 (see http://math.rice.edu/~dfield/dfpp.html ), as well as Maple (see http://www.maplesoft.com ) and lots of xfig. These notes are not copywrited by the author and any distribution of them is highly encouraged, especially without express written consent of the author. i ii Contents Lecture 1. Differential equations and their solutions . . . . . . . . . . . . . . 1 Lecture 2. First-order separable differential equations . . . . . . . . . . . . . 9 Lecture 3. Linear first-order differential equations . . . . . . . . . . . . . . . 15 Lecture 4. Applications of first-order differential equations . . . . . . . . . . 19 Lecture 5. Stability and phase plane analysis . . . . . . . . . . . . . . . . . . 27 Lecture 6. Exact differential equations . . . . . . . . . . . . . . . . . . . . . 31 Lecture 7. Substitutions for first-order differential equations . . . . . . . . . 37 Lecture 8. Second-order, constant-coefficient equations . . . . . . . . . . . . 43 Lecture 9. The Wronskian and linear independence . . . . . . . . . . . . . . 49 Lecture 10. Complex roots of the characteristic equation . . . . . . . . . . . 53 Lecture 11. Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Lecture 12. Nonhomogeneous equations: undetermined coefficients . . . . . . 61 Lecture 13. Nonhomogeneous equations: variation of parameters . . . . . . . 69 Lecture 14. Mechanical vibrations . . . . . . . . . . . . . . . . . . . . . . . . 75 Lecture 15. Forced mechanical vibrations . . . . . . . . . . . . . . . . . . . . 81 Lecture 16. Systems and linear algebra I: introduction . . . . . . . . . . . . 89 Lecture 17. Systems and linear algebra II: RREF . . . . . . . . . . . . . . . 97 Lecture 18. Systems and linear algebra III: linear dependence and independence103 Lecture 19. Systems and linear algebra IV: the inverse matrix and determinants107 Lecture 20. Systems and linear algebra V: eigenvalues and eigenvectors . . . 113 iii iv Lecture 1. Differential equations and their solutions 1. Algebraic equations An algebraic equation is an equation between an unknown quantity x and functions of this quantity x . It may be written in the form F ( x ) = 0 , such as 2 x 2 + x- 3 = 0 ....
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