Differential Equations

Announcements

★★★ 2025/08/23  ★★★ 本課程會搭配習題課，時間規劃為禮拜一晚上6-8點（一小時課本導讀、一小時練習解習題），同時於禮拜二課程結束也會佔用課後時間小考（請大家諒解課堂中考試會導致進度落後，不得不做此安排），目前時間規劃為課後11點半至12點間作為小考時段，請同學諒解並盡事先排開其他事務。

★★★ 2025/08/17  ★★★ 如果有意願選修這門課的同學，第一節課請務必出席，未出席者不論理由均不接受加簽！

★★★ 2025/08/06  ★★★ 課程網頁上線，目前細節暫定，請有意願修課的同學第一節課務必前來，以免損失個人權益。另外eCourse2僅會作為作業、筆記繳交、課程資料下載與緊急公告平台，課程規定與最新的詳細課程進度均會在本網頁公開，聯繫老師也請按照下述說明，請大家務必規律地到這裡查閱相關資訊！最後最後，微分方程並不難，難的通常是時間管理，大家可以讀讀來自學長姐的建議，在教學評鑑裡的課程回饋也有些可參考的學習方法，讓我們一起成長、蛻變吧 :)

Course Information

Instructor: Jian-Jia Weng

Time: Tuesdays and Thursdays, 10:15-11:30AM (C)

Location: R103, Innovation Building

Teaching Assistants:  聯絡助教請記得帶上禮貌，信件請同時郵件副本寄送給老師

• 陳怡鳳：yifengc3230 AT alum.ccu.edu.tw

• 鄭逸蓮：97052 AT alum.ccu.edu.tw

• 翁英訓：kevin930429 AT gmail.com

• 吳秉叡：haley040726@gmail.com

• 王柔淳：jouuuuu2002 AT gmail.com

• 楊日醇：anonymousunliquor AT gmail.com

• 郭亮榆：ajkunto0730 AT gmail.com

• 鄭丞恩：g13430016 AT ccu.edu.tw

Tutorial combined Office Hour: R101, Innovation Building; Mondays, 6:00-8:00PM

Teaching Method: Chalkboard Teaching

[Required] Textbook: Dennis G. Zill, Differential Equations with Boundary-Value Problems, 9th Ed|Metric Version, 2018, Cengage Learning

Grade Evaluation: 14 Quizzes (each 3.5%) + Midterm Exam (20%) + Final Exam (20%) + Notes&In-Class Participation (11%) 

Contact Information

Office: R428, Innovation Building  (please make an appointment before coming)

Campus Internal Phone Number：33528

Email: jjweng AT ccu.edu.tw

If you have any questions regarding the course, you can email me from your school email account with:  

• Subject: [DE 2025] Inquiry - Your name and Student ID number (example: [DE 2025] Inquiry - 周杰倫 1234567)

• Contents: (1) topics you want to discuss and (2) your preferred time to meet in person (please specify at least 3 time slots)

for a special accommodation. I should reply to your email within 24hours; if not, please send the email again.

Tentative Schedule

• Week 1 (09/09, 09/11): Pre-Quiz/Section 1.1

  1. Definition of Differential Equations

  2. Classifications of Differential Equations (ODE&PDE, Order, Linearity) 

  3. Definition of Solution(s) of an ODE (Interval of Definition, Explicit&Implicit Solutions)

  4. n-Paramater Family of Solutions/Particular Solution/General Solution(see P14 Rmk (iv))/Trivial Solution/Singular Solution

  5. Definitions of Initial Value Problem (IVP) and Boundary Value Problem (BVP)

本週重點：要成為一個微分方程的解有很多性質需滿足，譬如它必須要是可微分且微分後仍連續的函數（Why Why Why?

• Week 2 (09/16, 09/18): Quiz1/Sections 1.2-2.1.1

  1. The Existence and Uniqueness Theorem for 1st-Order ODE (Theorem 1.2.1)

  2. Construct a Solution Curve without a Solution: Direction Field of 1st-Order ODE (Section2.1.1)

  3. Euler's Method (Section 2.6)/Runge-Kunta Method (Section 9.x)

  4. Qualitative Analysis of Autonomous DE (Section 2.1.2)

  5. Separable ODE (Section 2.2, inspired by the integration by substitution)
     Recommended reading: Teaching integration by substitution by David Gale

本週重點1：原先有無窮多組解的1st-Order ODE會因initial conditions而變少(Ex4 of Sec.1.2)甚至無解
本週重點2：如何使用Theorem 1.2.1判斷一個1st-Order IVP是否有唯一解

• Week 3 (09/23, 09/25): Quiz2/Sections 2.1.2-2.4

  1. Linear ODE (Section 2.3, inspired by the product rule of derivative)

  2. Exact ODE (Section 2.4, inspired by the fact that an implicit solution induces a DE)

本週重點：求解分離型、線性型、正合型一階ODE的想法以及熟悉使用這三種Methods of Solution

• Week 4 (09/30, 10/02): Quiz3/Sections 2.4-2.5

  1. Review of 1st-Order Linear ODE and Exact ODE

  2. Convert Non-excat ODE into Exact ODE

  3. Solutions by Substitutions (Section 2.5, Type I: f(Ax+By+c) and homogeneous DE)

本週重點1：如何將non-Exact ODE轉換成Exact ODE
本週重點2：如何利用變數變換將ODE轉換成分離或線性型 ODE

• Week 5 (10/07, 10/09): Quiz4/Sections 2.5, 4.1

  1. Solutions by Substitutions (Section 2.5, Bernoulli's Equation)

  2. The Linear Algebra behind Linear ODE 

  3. Linear Operator (acts on continuously differentiable functions

  4. Linearly Dependence and Wroskian

本週重點1：線性代數求解聯立方程組與線性微分方程求解之關係，針對齊次線性微分方程，求解過程是在找一個線性微分算子的核(Kernel)
本週重點2：函數線性相依的定義以及如何使用行列式來判斷函數間是否線性相依

• Week 6 (10/14, 10/16): Quiz5/Sections 4.1, 4.2 

  1. Review of Week 5

  2. Existence and Uniqueness Theorem for Linear IVP 

  3. Homogeneous vs Non-Homogeneous

  4. Polynomial Differential Operator

  5. Reduction of Order 

本週重點1：如何確定線性微分方程的IVP之解存在與唯一
本週重點2：線性ODE的解之基礎特性，通解 =  特解 +  齊次解，其中齊次解包含線性獨立解
本週重點3：如何從已知的部分（齊次）解找出其他（齊次）解

• Week 7 (10/21, 10/23): Quiz6/Sections 4.3

  1. n-th Order Homogeneous Linear ODE with Constant Coefficients

  2. Finding Particular Solutions
     
     

本週重點1：學習求解高階齊次線性常係數微分方程的方法以及講解課本沒說明的「公式」

本週重點2：利用系統觀點講述上述兩點作法的原理！

• Week 8 (10/28, 10/30): Quiz7/Sections 4.4-4.5, 4.7

  1. Finding Particular Solutions via Superposition Principle

  2. Finding Particular Solutions via Annihilator Approach

  3. (Homogeneous) Cauchy-Euler Equation

  4. Variation of Parameter

本週重點：

• Week 9 (11/04, 11/06): Quiz8/Official Midterm Exam Week/Sections 4.6, 4.9, 4.10

  1. Elimination of Linear ODE System

  2. Non-linear ODEs

  3. Mathematical Modeling of Higher Order ODEs

  4. Spring-Mass Systems (Free/Forced; Undamped/Damped)

  5. Catenary Problem （懸鍊線問題）

本週重點1：可以利用Cramer's rule的方法來求微分方程組的解，超酷！

本週重點2：在微分方程裡也有類似高斯消去法的方法可以幫助求解微分方程！

• Week 10 (11/11, 11/13): 

  1. 
     

本週重點：

• Week 11 (11/18, 11/20): Quiz9/Sections 5.1, 5.3, 6.1

  1. Power Series

  2. Ordinary and Singular Points

  3. Solution about Ordinary Points

本週重點：針對Ordinary Points展演如何利用冪級數來求解微分方程

• Week 12 (11/25, 11/27): Quiz10/Sections 6.2-6.3

  1. Regular and Irregular Singular Points

  2. Solution about Singular Points

  3. (Unilateral) Laplace Transform (for functions defined on [0, \infty))

本週重點1：Ordinary及Regular Singular Points的解形式、Indicial Equation的來由以及它在求解過程扮演的角色
本週重點2：什麽是單邊拉氏轉換（需注意我們處理的函數定義域為t>=0，即Causal Function）以及它的物理意義
本週重點3：習題課詳述Frobenius Method的必要性和源由，包含前人是如何得到那三個狀況的解形式

• Week 13 (12/02, 12/04): Quiz11/Sections 7.1-7.3

  1. Basic Laplace Transform and Inverse Transform

  2. The Laplace Transform of derivatives of functions 

本週重點1：拉氏轉換的緣由（傅立葉轉換無法處理的函數可藉由引入適當衰退項）以及一些基本函數的拉式轉換結果，其中拉式轉換是線性運算
本週重點2：如何利用部分分式分解(Partial Fraction Decomposition)以及查表找Inverse Laplace（注意逆拉式轉換也是線性運算）

• Week 14 (12/09, 12/11): Quiz12/Sections 7.3  

  1. Application of Laplace Transform to Solve IVPs

  2. Laplace Transform Properties (shift in t- or s-axis and scaling)

  3. Derivatives of Laplace Transform（在s-domain上微分，在t-domain上會發生什麽事呢？

  4. Integral of Laplace Transform（在s-domain上積分，在t-domain上會發生什麽事呢？

  5. Convolution Theorem（我們因為有規範causal function這個特性，這公式跟未來你在訊號與系統看到的形式會有一點點差異

  6. 
     

本週重點：在s-domain/t-domain上做座標的平移/縮放會造成哪些影響。如何使用性質以及Convolution Theorem來擴增拉氏轉換對。

• Week 15 (12/16, 12/18): Quiz13/Sections 7.4

  1. Volterra Integral Equation

  2. Laplace Transform of Periodic Functions

  3. Dirac Delta Function

  4. Solving a Linear System of ODEs via Laplace Transform

  5. The Origin of Fourier Series (Heat Equation)

本週重點：

• Week 16 (12/23, 12/25): Official Final Exam Week, Sections 7.5-7.6 and Section 11

  1. The Origin of Fourier Series (Cont'd)

  2. Orthonormal Set of Functions

  3. Fourier Sine/Cosine Series

本週重點：

• Week 17 (12/30, 01/01)：
  
  

• Week 18 (01/06, 01/08): 考後課程總結

  1. Partial Differential Equations (PDE)

  2. Separation of Variables (for PDE)

  3. Wave Equation

  4. Laplace Equation

本週重點：

Selected Homework Problems

You do not need to submit answers to any homework problems. You are encouraged to work them together with your friends to save your time .   

• Section 1: 

  • 1-1: 1, 3, 5, 9, 12, 15, 23, 27, 29, 30, 31, 33, 36, 43, 45, 46, 47, 53

  • 1-2: 3, 4, 11, 12, 21, 25, 31, 36, 50*

  • 1-3: 3, 15, 17, 21

• Section 2: 

  • 2-1: 1, 12, 21, 33*, 34*

  • 2-2: 2, 11, 28, 31, 40, 52*

  • 2-3: 5, 12, 21, 32, 49*, 50*, 51*, 52*, 53*

  • 2-4: 1, 6, 7, 9, 11, 13, 15, 19, 26, 27, 33, 35, 37, 39

  • 2-5: 1, 7, 9, 13, 17, 18, 19, 21, 25, 27, 29, 35

  • 2-6: 3

• Section 3: 

  • 3-1: 3, 19, 27, 33, 35, 39, 43, 45

  • 3-2: 5, 7, 9, 11, 15, 19, 21

• Section 4: 

  • 4-1: 1, 9, 13, 17, 21, 25, 35

  • 4-2: 1, 3, 13, 17, 19

  • 4-3: 3, 5, 17, 21, 25, 31, 37, 51, 57

  • 4-4: 1, 7, 13, 27, 29, 35

  • 4-5: 15, 21, 25, 49, 65, 69, 71

  • 4-6: 1, 3, 5, 9, 17, 21, 25, 27

  • 4-7: 1, 5, 15, 21, 25, 29, 31, 35, 41

  • 4-8: 

  • 4-9: 1, 7, 11, 15, 19, 21

  • 4-10: 3, 5, 7, 9, 13, 19

• Section 5:

  • 5-1: 1, 7, 13, 21, 35, 49, 57

  • 5-3: 7, 15, 17, 19 

• Section 6:

  • 6-1: 1, 7, 13, 15, 19, 23, 25, 29, 35

  • 6-2: 1, 3, 13, 15, 19, 21, 23

  • 6-3: 1, 3, 5, 9, 11, 13, 17, 25, 27, 29, 33

• Section 7:

  • 7-1: 3, 5, 13, 15, 29, 35, 43, 50, 53, 54, 55

  • 7-2: 1, 3, 13, 15, 19, 21, 29, 35, 43

  • 7-3: 9, 17, 19, 25, 31, 39, 47, 49, 51, 55, 59, 65, 69, 83

  • 7-4: 5, 13, 17, 23, 29, 39, 49, 51, 53, 63, 66, 67

  • 7-5: 5, 11, 13

  • 7-6: 7, 11, 15, 17

• Section 11:

  • 11-1: 3, 5, 7, 17, 22

  • 11-2: 5, 7, 9, 13, 19, 21, 23

  • 11-3: 7, 13, 19, 23, 27, 31, 37, 45

• Section 12:

  • 12-1: 9. 15, 17, 22

  • 12-2: 1, 3, 7, 11

  • 12-4: 3, 7, 9, 11, 14

  • 12-5: 5, 7, 12, 15, 19

Reading Assignments

• Week 0: Sections 1.1-1.2

• Week 1: Sections 1.3-2.1

• Week 2: Sections 2.2-2.4

• Week 3: Sections 2.5-2.6

• Week 4: Sections 3.1-3.2

• Week 5: Section 4.1

• Week 6: Section 4.2-4.3

• Week 7: Sections 4.4, 4.7

• Week 8: Sections 4.5-4.6

• Week 9: Sections 4.8-4.10

• Week 10: Sections 5.1, 5.3

• Week 11: Sections 6.1-6.3

• Week 12: Sections 7.1-7.3

• Week 13: Sections 

• Week 14: Sections 

• Week 15: Section 

• Week 16: Section 

• Week 17: Section 

• Week 18: Section 

References

1. P. Blanchard, R. L. Devaney, and G. R. Hall, Differential Equations, 4ed, International Edition, Brooks/COLE: Cengage Learning, 2011.（東華書局|新月圖書代理）

2. C. H. Edwards and D. E. Penny, Differential Equations, Computing and Modeling, 4ed, Pearson Education, 2008, ISBN-13: 9780136004387

Applications in Communication Systems

1. K. K. Leung, W. A. Massey and W. Whitt, "Traffic models for wireless communication networks," IEEE J. Sel. Areas Commun., vol. 12, no. 8, pp. 1353-1364, Oct. 1994, doi: 10.1109/49.329340.

2. M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi and D. A. Spielman, "Efficient erasure correcting codes,"  IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 569-584, Feb 2001, doi: 10.1109/18.910575.

3. L. Wang, G. Liu, J. Xue and K. -K. Wong, "Channel prediction using ordinary differential equations for MIMO Systems," IEEE Trans. Veh. Tech., vol. 72, no. 2, pp. 2111-2119, Feb. 2023, doi: 10.1109/TVT.2022.3211661. 

4. N.-K. Ayano and T. Wadayama, "Ordinary differential equation-based MIMO signal detection." arXiv preprint arXiv:2304.14097 (2023).