Differential Equations (DE) 工程數學--微分方程 Differential Equations (DE) 授課者：丁建均 教學網頁：http://djj.ee.ntu.edu.tw/DE.htm (請上課前來這個網站將講義印好) 歡迎大家來修課！

授課者：丁建均 Office： 明達館723室, TEL： 33669652 Office hour： 星期三下午 1:00~5:00 個人網頁：http://disp.ee.ntu.edu.tw/          E-mail:  djj@cc.ee.ntu.edu.tw,    djj1@ms63.hinet.net 上課時間： 星期三 第 3, 4 節 (AM 10:20~12:10)   星期五 第 2 節 (AM 9:10~10:00) 上課地點： 電二143 課本： "Differential Equations-with Boundary-Value Problem", 7th edition, Dennis G. Zill and Michael R. Cullen 評分方式：四次作業一次小考 10%,   期中考 45%,  期末考 45%    

注意事項： 請上課前，來這個網頁，將上課資料印好。 http://djj.ee.ntu.edu.tw/DE.htm (2) 請各位同學踴躍出席 。 (3) 作業不可以抄襲。作業若寫錯但有用心寫仍可以有40%~90% 的分數，但抄襲或借人抄襲不給分。 (4) 我週一至週四下午都在辦公室，有什麼問題 ，歡迎同學們來找我

上課日期 Date (Wednesday, Friday) Remark 1. 9/12, 9/14 2. 9/19, 9/21 3.   2. 9/19, 9/21 3. 9/26, 9/28 4. 10/3, 10/5 5. 10/12 10/10 國慶 6. 10/17, 10/19 7. 10/24, 10/26 8. 10/31, 11/2 9. 11/7: Midterm; (Chaps.1-5), 11/9 範圍： (Chaps.1-5) 10. 11/14, 11/16 11. 11/21, 11/23 12. 11/28, 11/30 13. 12/5, 12/7 14. 12/12, 12/14   15. 12/19, 12/21 16. 12/26, 12/28 17. 1/2, 1/4 18. 1/9 Finals  範圍： (Chaps. 6, 7, 11, 12, 14)

課程大綱 Introduction (Chap. 1) 解法 (Chap. 2) First Order DE 應用 (Chap. 3) Higher Order DE 應用 (Chap. 5) 多項式解法 (Chap. 6) Partial DE (Chap. 12) Laplace Transform (Chap. 7) Transforms Fourier Series (Chap. 11) Fourier Transform (Chap. 14)

Chapter 1 Introduction to Differential Equations 1.1 Definitions and Terminology (術語) Differential Equation (DE): any equation containing derivation (page 2, definition 1.1) x: independent variable 自變數 y(x): dependent variable 應變數

Note: In the text book, f(x) is often simplified as f notations of differentiation , , , , ………. Leibniz notation , , , , ………. prime notation , , , , ………. dot notation , , , , ………. subscript notation

(2) Ordinary Differential Equation (ODE): differentiation with respect to one independent variable (3) Partial Differential Equation (PDE): differentiation with respect to two or more independent variables

(4) Order of a Differentiation Equation: the order of the highest derivative in the equation 7th order 2nd order

(5) Linear Differentiation Equation: All the coefficient terms are independent of y. Property of linear differentiation equations: If and y3 = by1 + cy2, then

(6) Non-Linear Differentiation Equation

(7) Explicit Solution (page 6) The solution is expressed as y = (x) (8) Implicit Solution (page 7) Example: , Solution: (implicit solution) or (explicit solution)

1.2 Initial Value Problem (IVP) A differentiation equation always has more than one solution. for , y = x, y = x+1 , y = x+2 … are all the solutions of the above differentiation equation. General form of the solution: y = x+ c, where c is any constant. The initial value (未必在 x = 0) is helpful for obtain the unique solution. and y(0) = 2 y = x+2 and y(2) =3.5 y = x+1.5

The kth order differential equation usually requires k initial conditions (or k boundary conditions) to obtain the unique solution. solution: y = x2/2 + bx + c, b and c can be any constant y(1) = 2 and y(2) = 3 y(0) = 1 and y'(0) =5 y(0) = 1 and y'(3) =2 For the kth order differential equation, the initial conditions can be 0th ~ (k–1)th derivatives at some points. (boundary conditions，在不同點) (initial conditions) (boundary conditions，在不同點)

1.3 Differential Equations as Mathematical Model Physical meaning of differentiation: the variation at certain time or certain place Example 1: A: population 人口增加量和人口呈正比

Example 2: T: 熱開水溫度, Tm: 環境溫度 t: 時間

大一微積分所學的： 的解 例如： 問題： (1) 若等號兩邊都出現 dependent variable (如 pages 15, 16 的例子) (2) 若order of DE 大於 1

DE Review dependent variable and independent variable PDE and ODE Order of DE linear DE and nonlinear DE explicit solution and implicit solution initial value IVP

Chapter 2 First Order Differential Equation 2-1 Solution Curves without a Solution Instead of using analytic methods, the DE can be solved by graphs (圖解) slopes and the field directions: x-axis (x0, y0) the slope is f(x0, y0)

Example 1 dy/dx = 0.2xy 資料來源： Fig. 2-1-3(a) of “Differential Equations-with Boundary-Value Problem”, 7th ed., Dennis G. Zill and Michael R. Cullen.

Example 2 dy/dx = sin(y), y(0) = –3/2 With initial conditions, one curve can be obtained 資料來源： Fig. 2-1-4 of “Differential Equations-with Boundary-Value Problem”, 7th ed., Dennis G. Zill and Michael R. Cullen.

Advantage: It can solve some 1st order DEs that cannot be solved by mathematics. Disadvantage: It can only be used for the case of the 1st order DE. It requires a lot of time

Section 2-6 A Numerical Method Another way to solve the DE without analytic methods independent variable x x0, x1, x2, ………… Find the solution of Since approximation sampling(取樣) 前一點的值 取樣間格

Example: dy(x)/dx = 0.2xy y(xn+1) = y(xn) + 0.2xn y(xn )*(xn+1 –xn). dy/dx = sin(x) y(xn+1) = y(xn) + sin(xn)*(xn+1 –xn). . 後頁為 dy/dx = sin(x), y(0) = –1, (a) xn+1 –xn = 0.01, (b) xn+1 –xn = 0.1, (c) xn+1 –xn = 1, (d) xn+1 –xn = 0.1, dy/dx = 10sin(10x) 的例子 Constraint for obtaining accurate results: (1) small sampling interval (2) small variation of f(x, y)

(a) (b) (c) (d)

Advantages -- It can solve some 1st order DEs that cannot be solved by mathematics. -- can be used for solving a complicated DE (not constrained for the 1st order case) -- suitable for computer simulation Disadvantages -- numerical error (數值方法的課程對此有詳細探討)

Exercises for Practicing (not homework, but are encouraged to practice) 1-1: 1, 13, 19, 23, 33 1-2: 3, 13, 21, 33 1-3: 2, 7, 28 2-1: 1, 13, 20, 25, 33 2-6: 1, 3

頁碼 作品 版權圖示 來源/作者 台灣大學 電信工程研究所 丁建均教授 以創用CC「姓名標示－非商業性－相同方式分享」臺灣3.0版授權釋出。 19 台灣大學 電信工程研究所 丁建均教授 以創用CC「姓名標示－非商業性－相同方式分享」臺灣3.0版授權釋出。 20 21 25