Engineering Mathematics Instructor : Prof. Sei-Wang Chen, PhD

Teaching Goal: Study on Engineering Mathematics that is often required in various scientific areas. This course focuses on practical techniques and provides detailed steps for solving differential equations, systems of linear differential equations, and vector calculus.

Advanced Engineering Mathematics 6th Edition, Concise Edition Textbook: Advanced Engineering Mathematics 6th Edition, Concise Edition by Peter V. O’Neil (2007) Bookstore: Thomson Books/Cole, U. S. A. 華泰文化, TEL: 21621217 程元杰, 0910385337

Contents of Textbook Part1: Chapters 1, 2, 3 Part2: Chapters 4, 5, 6, 7 Part3: Chapters 8 Part4: Chapters 9, 10 Part5: Chapters 11, 12, 13 Part6: Chapters 14, 15, 16

Chapter 1 : First-Order Differential Equations Chapter 2 : Second-Order Differential Chapter 3 : The Laplace Transform Chapter 4 : Series Solutions Chapter 8 : Systems of Linear Differential Chapter 9 : Vector Differential Calculus Chapter 10: Vector Integral Calculus Find the power points of the chapters by CSIE --> SW Chen --> Teaching --> Engineering mathematics

Syllabus Week Content 1 Ch1 2 Ch1 3 Ch2 4 Ch2 5 Ch3 6 Ch3 7 Ch3 8 Mid-Term Exam

Week Content 9 Ch8 10 Ch8 11 Ch8 12 Ch9 13 Ch9 14 Ch9 15 Ch10 16 Ch10 17 Ch10 18 Final Exam (or present)

參考書目: Advanced Engineering Mathematics, by R. J. Lopez, 2001, Addison Wesley, New York, U.S.A., 2001, 新月圖書公司, 23317856.

Submit Assignments Oct. 6 (四) 停課一次 只有作業,沒考試 (1) Prepare a file: File name: StudentID_Name_HW#.doc e.g., 699470486S_郭彥佑_HW1. doc (2) Submit the file to: ftp://140.122.184.4:9876 User ID: EM100; Password: EM100 Oct. 6 (四) 停課一次 只有作業,沒考試

Late homework and reports will not be accepted Evaluation: Examination I 25% Examination II 25% Homework 35% Attendance 15% Late homework and reports will not be accepted Plagiarism is definitely not allowed

Ordinary Differential Equations Part 1: Ordinary Differential Equations Ch1: First-Order Differential Equations Ch2: Second-Order Differential Equations Ch3: The Laplace Transform Ch4: Series Solutions

Differential equation --- contains derivatives e.g., or or y : function of x, x : independent variable (b) Ordinary differential equation – involves only total derivatives Partial differential equation – involves partial derivatives What is

(c) The order of a differential equation e. g (c) The order of a differential equation e.g., ○ The solution of a differential equation is a function y(x) of independent variable x that may be defined on e.g., i) , solution: y = sin2x for ii) , solution: y = xlnx – x for x > 0

Ch. 1: First-Order Differential Equations 1.1. Preliminary Concepts ○ First-order differential equation: -- involves a first but no higher derivatives e.g., y：function of x x：independent variable : solution

1.1.1. General and Particular Solutions ○ General solution: arbitrary constant Substitute into (A) Particular solutions: k = 1, ; k = 2 , k = ,

1.1.2. Implicitly Defined Solutions ○ Explicit function: Implicit function: e.g., , ○ (Explicit solution) (Implicit solution)

1.1.3. Integral Curves ○ Example 1.1: General solution: -- Help to comprehend the behavior of solution ○ Example 1.1: General solution:

1.1.4. Initial Value Problems ○ initial condition Graphically, the particular integral curve passes through point ( ) The objective is to obtain a unique solution ○ Example 1.4: , initial condition General solution: ,

-- A set of line segments tangent to a curve 1.1.5. Direction Field -- A set of line segments tangent to a curve -- Give a rough outline of the shape of the curve ○ Giving , instead of solving for y, solving for

-- A set of line segments tangent to a curve 1.1.5. Direction Field -- A set of line segments tangent to a curve -- Give a rough outline of the shape of the curve ○ Giving , instead of solving for y, solving for 20

Slope: , General Solution: ○ Example 1.5: Slope: , General Solution: Figure 1.5. Direction field for y'= y² and integral curves through (0,1), (0,2), (0,3), - 1, (0, - 2), and (0, - 3).

1.2. Separable Equations

○ Example 1.7: , (the general solution) y = 0 is a solution, called a singular solution, it cannot be obtained from the general solution

1.3. Linear Differential Equations -------- (A) (1) Find integrating factor: (2) Multiply (A) by (3)

○ Example 1.14: i) Integrating factor ii) Multiply the equation by , iii) iv) Integrate

1.4. Exact Differential Equations can be written as ------- (A) If , s.t. and (implicitly define the solution) : potential function

○ Example 1.17: Let From

＊ We can start with , then as well From Solution: ＊ We can start with , then as well 28

c(y) is not independent of x No potential function ＊ Not every is exact e.g. From c(y) is not independent of x No potential function

Theorem 1.1: Exactness is exact iff (a) If is exact, then s.t. (b) If , show s.t.

Let : any point

○ Example : 1.5. Integrating Factors If : not exact But : exact is not exact 1.5. Integrating Factors If : not exact But : exact integrating factor

. ○ How to find : exact, Try as , or ○ Example 1.21: The equation is not exact

Consider Let Try (B) Integrate

(A) Let be the potential function The implicit solution: The explicit solution:

○ Example 1.22: Let Find integrating factor by

Try This cannot be solved for as a function of x (ii) Likewise, let

(iii) Try (B) Divide by

: independent Multiply (A) by Let Form

From Obtain the potential function The implicit solution: or

1.6. Homogeneous, Bernoulli, and Riccati Eqs. 1.6.1 Homogeneous Equation: ----- (A) ＊ A homogeneous equation is always transformed into a separable one by letting ↑=1 (A)

○ Example 1.25: Let (A)

1.6.2. Bernoulli Equation linear separable can be transformed into linear by

○ Example 1.27: Let (A) Multiply by (linear) Integrating factor:

1.6.3. Riccati Equation Let S(x) be a solution and let The Riccati equation is transformed into linear ○ Example 1.28: By inspection, is a solution of (A) Let

(linear) Integrating factor: Integrate and Solution:

Initial Value Problems: 1.8. Existence and Uniqueness for Solutions of Initial Value Problems Initial Value Problems: The problem may have no solution and may have multiple solutions ○ Example 1.30: The equation is separable and has solution The equation has no real solution. is not a solution because it does not satisfy the initial condition.

○ Example 1.31: The equation has solution This problem has multiple solutions i, Trivial solution: ii, Define Consider All satisfy the initial condition

○ Theorem 1.2： If f, : continuous in , then s.t. The initial value problem has a unique solution defined on The size of h depends on f and

the entire plane and hence on ○ Example 1.31: not continuous on (x, 0) ○ Example：The problem : both continuous on the entire plane and hence on s.t. the problem has a unique solution in Solve the problem , which is valid for , we can take 50

○ Theorem 1.3： : continuous on I The problem , has a unique solution defined Proof： From the general solution of the linear equation and the initial condition , the solution of the initial value problem is : continuous on I, the solution is defined

Homework 1 Chapter 1 Sec.1.1: 1, 2, 7, 12 Sec.1.2: 1, 2, 11 Sec.1.3: 1 Sec.1.4: 1 Sec.1.5: 1

1.5(1): Determine a test involving M and N to tell when has an integrating factor that is a function of y only. Ans: Let be an integrating factor such that is exact. Then,

must be independent of x. The test is then that must be independent of x. 54

Homework 2 Chapter 1 Sec.1.6: 15, 16, 20, 21 Sec.1.8: 1, 3, 5

Chapter 2: Second-Order Differential Equations 2.1. Preliminary Concepts ○ Second-order differential equation e.g., Solution: A function satisfies , (I : an interval)

○ Linear second-order differential equation Nonlinear: e.g., 2.2. Theory of Solution ○ Consider y contains two parameters c and d

The graph of Given the initial condition

Given another initial condition The graph of ◎ The initial value problem: ○ Theorem 2.1: : continuous on I, has a unique solution

○ Theorem 2.2: : solutions of Eq. (2.2) 2.2.1.Homogeous Equation ○ Theorem 2.2: : solutions of Eq. (2.2) solution of Eq. (2.2) : real numbers Proof:

※ Two solutions are linearly independent. Their linear combination provides an infinity of new solutions ○ Definition 2.1: f , g : linearly dependent If s.t. or ; otherwise f , g : linearly independent In other words, f and g are linearly dependent only if for

○ Wronskian test -- Test whether two solutions of a homogeneous differential equation are linearly independent Define: Wronskian of solutions to be the 2 by 2 determinant

○ Let If : linear dep., then or Assume

○ Theorem 2.3: 1) Either or 2) : linearly independent iff Proof (2): (i) (if : linear indep. (P), then (Q) if ( Q) , then : linear dep. ( P) ) : linear dep.

(ii) (if (P), then : linear indep. (Q) if : linear dep. ( Q), then ( P)) : linear dep., ※ Test at just one point of I to determine linear dependency of the solutions

。 Example 2.2: are solutions of : linearly independent

。 Example 2.3: Solve by a power series method The Wronskian of at nonzero x would be difficult to evaluate, but at x = 0 are linearly independent

○ Definition 2.2: ◎ Find all solutions 1. : linearly independent : fundamental set of solutions 2. : general solution : constant ○ Theorem 2.4: : linearly independent solutions on I Any solution is a linear combination of

Proof: Let be a solution. Show s.t. Let and Then, is the unique solution on I of the initial value problem

2. 2. 2. Nonhomogeneous Equation ○ Theorem 2 2.2.2. Nonhomogeneous Equation ○ Theorem 2.5: : linearly independent homogeneous solutions of : a nonhomogeneous solution of any solution has the form

Proof: Given , solutions : a homogenous solution of : linearly independent homogenous solutions (Theorem 2.4)

1. Find the general homogeneous solutions ○ Steps: 1. Find the general homogeneous solutions of 2. Find any nonhomogeneous solution of 3. The general solution of is 2.3. Reduction of Order -- A method for finding the second independent homogeneous solution when given the first one

○ Let Substituting into ( : a homogeneous solution ) Let (separable)

For symlicity, let c = 1, 。 Example 2.4: : a solution Let : independent solutions 。 Example 2.4: : a solution Let

Substituting into (A), For simplicity, take c = 1, d = 0 : independent The general solution:

2. 4. Constant Coefficient Homogeneous A, B : numbers ----- (2 2.4. Constant Coefficient Homogeneous A, B : numbers ----- (2.4) The derivative of is a constant (i.e., ) multiple of Constant multiples of derivatives of y , which has form , must sum to 0 for (2,4) ○ Let Substituting into (2,4), (characteristic equation)

i) Solutions : : linearly independent The general solution:

。 Example 2.6: Let , Then Substituting into (A), The characteristic equation: The general solution:

ii) By the reduction of order method, Let Substituting into (2.4)

Choose : linearly independent The general sol. : 。 Example 2 Choose : linearly independent The general sol.: 。 Example 2.7: Characteristic eq. : The repeated root: The general solution:

iii) Let The general sol.:

。 Example 2.8: Characteristic equation: Roots: The general solution: ○ Find the real-valued general solution 。 Euler’s formula:

Maclaurin expansions:

。 Eq. (2.5),

Find any two independent solutions Take The general sol.:

2.5. Euler’s Equation , A , B : constants -----(2.7) Transform (2.7) to a constant coefficient equation by letting

Substituting into Eq. (2. 7), i. e. , --------(2 Substituting into Eq. (2.7), i.e., --------(2.8) Steps: (1) Solve (2) Substitute (3) Obtain

Characteristic equation: Roots: General solution: 。 Example 2.11: ------(A) -------(B) (i) Let Substituting into (A) Characteristic equation: Roots: General solution:

○ Solutions of constant coefficient linear equation have the forms: Solutions of Euler’s equation have the forms:

2.6. Nonhomogeneous Linear Equation ------(2.9) The general solution: ◎ Two methods for finding (1) Variation of parameters -- Replace with in the general homogeneous solution Let Assume ------(2.10) Compute

Substituting into (2.9), -----------(2.11) Solve (2.10) and (2.11) for Likewise,

。 Example 2. 15: ------(A) i) General homogeneous solution : Let 。 Example 2.15: ------(A) i) General homogeneous solution : Let . Substitute into (A) The characteristic equation: Complex solutions: Real solutions: :independent

ii) Nonhomogeneous solution Let

iii) The general solution:

(2) Undetermined coefficients Apply to A, B: constants Guess the form of from that of R e.g. : a polynomial Try a polynomial for : an exponential for Try an exponential for

。 Example 2.19: ---(A) It’s derivatives can be multiples of or Try Compute Substituting into (A),

: linearly independent and The homogeneous solutions: The general solution:

。 Example 2. 20: ------(A) , try Substituting into (A), 。 Example 2.20: ------(A) , try Substituting into (A), * This is because the guessed contains a homogeneous solution Strategy: If a homogeneous solution appears in any term of , multiply this term by x. If the modified term still occurs in a homogeneous solution, multiply by x again

Try Substituting into (A),

○ Steps of undetermined coefficients: (1) Find homogeneous solutions (2) From R(x), guess the form of If a homogeneous solution appears in any term of , multiply this term by x. If the modified term still occurs in a homogeneous solution, multiply by x again (3) Substitute the resultant into and solve for its coefficients

○ Guess from Let : a given polynomial , : polynomials with unknown coefficients Guessed

2.6.3. Superposition Let be a solution of is a solution of (A)

。 Example 2.25: The general solution: where homogeneous solutions

Chapter 3: The Laplace Transform 3.1. Definition and Basic Properties 。 Objective of Laplace transform -- Convert differential into algebraic equations ○ Definition 3.1: Laplace transform s.t. converges s, t : independent variables ＊ Representation:

。Example 3.2: Consider

Not every function has a Laplace transform * Not every function has a Laplace transform. In general, can not converge 。Example 3.1:

○ Definition 3.2.: Piecewise continuity (PC) f is PC on if there are finite points s.t. and are finite i.e., f is continuous on [a, b] except at finite points, at each of which f has finite one-sided limits

If f is PC on [0, k], then so is and exists

◎ Theorem 3.2: Existence of f is PC on If Proof:

* Theorem 3.2 is a sufficient but not a necessary condition.

＊ There may be different functions whose Laplace transforms are the same e.g., and have the same Laplace transform ○ Theorem 3.3: Lerch’s Theorem ＊ Table 3.1 lists Laplace transforms of functions

○ Theorem 3. 1: Laplace transform is linear Proof: ○ Definition 3. 3: ○ Theorem 3.1: Laplace transform is linear Proof: ○ Definition 3.3:. Inverse Laplace transform e.g., ＊ Inverse Laplace transform is linear

3.2 Solution of Initial Value Problems Using Laplace Transform ○ Theorem 3.5: Laplace transform of f: continuous on : PC on [0, k] Then, ------(3.1)

Proof: Let

○ Theorem 3.6: Laplace transform of : PC on [0, k] for s > 0, j = 1,2 … , n-1

。 Example 3.3: From Table 3.1, entries (5) and (8)

○ Laplace Transform of Integral From Eq. (3.1),

3.3. Shifting Theorems and Heaviside Function 3.3.1.The First Shifting Theorem ◎ Theorem 3.7: ○ Example 3.6: Given

○ Example 3.8:

3.3.2. Heaviside Function and Pulses ○ f has a jump discontinuity at a, if exist and are finite but unequal ○ Definition 3.4: Heaviside function 。 Shifting

。 Laplace transform of heaviside function

3.3.3 The Second Shifting Theorem Proof:

○ Example 3.11: Rewrite

◎ The inverse version of the second shifting theorem ○ Example 3.13: where rewritten as

3.4. Convolution

◎ Theorem 3.9: Convolution theorem Proof:

◎ Theorem 3.10: ○ Exmaple 3.18 ◎ Theorem 3.11: Proof :

○ Example 3.19:

3.5 Impulses and Dirac Delta Function ○ Definition 3.5: Pulse ○ Impulse: ○ Dirac delta function: A pulse of infinite magnitude over an infinitely short duration

○ Laplace transform of the delta function ◎ Filtering (Sampling) ○ Theorem 3.12: f : integrable and continuous at a

Proof:

by Hospital’s rule ○ Example 3.20:

3.6 Laplace Transform Solution of Systems ○ Example 3.22 Laplace transform Solve for

Partial fractions decomposition Inverse Laplace transform

3.7. Differential Equations with Polynomial Coefficient ◎ Theorem 3.13: Proof: ○ Corollary 3.1:

○ Example 3.25: Laplace transform

Find the integrating factor, Multiply (B) by the integrating factor

Inverse Laplace transform

○ Apply Laplace transform to algebraic expression for Y Differential equation for Y

◎ Theorem 3.14: PC on [0, k],

○ Example 3.26: Laplace transform ------(A) ------(B)

Finding an integrating factor, Multiply (B) by ,

In order to have

Formulas: ○ Laplace Transform: ○ Laplace Transform of Derivatives: ○ Laplace Transform of Integral:

○Shifting Theorems: ○ Convolution: Convolution Theorem: ○

○