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

2. 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.

3. 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:
程元杰,

4. 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

5. 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

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

7. Week Content
Ch8
Ch8
Ch8
Ch9
Ch9
Ch9
Ch10
Ch10
Ch10
Final Exam (or present)

8. 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,

9. 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., S_郭彥佑_HW1. doc (2) Submit the file to: ftp:// :9876 User ID: EM100; Password: EM100
Oct. 6 (四) 停課一次
只有作業,沒考試

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

11. 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

12. 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

13. (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

14. 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

15. 1.1.1. General and Particular Solutions○
General solution:
arbitrary constant
Substitute into (A)
Particular solutions:
k = 1, ; k = 2 ,
k = ,

16. 1.1.2. Implicitly Defined Solutions
○ Explicit function:
Implicit function:
e.g., , ○
(Explicit solution)
(Implicit solution)

17. 1.1.3. Integral Curves ○ Example 1.1: General solution:
-- Help to comprehend the behavior of solution
○ Example 1.1:
General solution:

18. 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: ,

19. -- A set of line segments tangent to a curve
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. -- A set of line segments tangent to a curve
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

21. 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).

22. 1.2. Separable Equations

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

24. 1.3. Linear Differential Equations -------- (A)
(1) Find integrating factor:
(2) Multiply (A) by
(3)

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

26. 1.4. Exact Differential Equations
can be written as
(A)
If , s.t.
and
(implicitly define the solution)
: potential function

27. ○ Example 1.17: Let From

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

29. 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

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

31. Let
: any point

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

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

34. Consider Let Try (B) Integrate

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

36. ○ Example 1.22:
Let
Find integrating factor by

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

38. (iii) Try (B) Divide by

39. : independent
Multiply (A) by
Let
Form

40. From Obtain the potential function The implicit solution: or

41. 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)

42. ○ Example 1.25: Let (A)

43. 1.6.2. Bernoulli Equation linear separable
can be transformed into linear by

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

45. 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

46. (linear)
Integrating factor:
Integrate and
Solution:

47. 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.

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

49. ○ 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

50. 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

51. ○ 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

52. 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

53. 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,

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

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

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

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

58. The graph of Given the initial condition

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

60. ○ 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:

61. ※ 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

62. ○ 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

63. ○ Let If : linear dep., then or Assume

64. ○ 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.

65. (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

66. 。 Example 2.2:
are solutions of
: linearly independent

67. 。 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

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

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

70. 2. 2. 2. Nonhomogeneous Equation ○ Theorem 2
Nonhomogeneous Equation ○ Theorem 2.5: : linearly independent homogeneous solutions of : a nonhomogeneous solution of any solution has the form

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

72. 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

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

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

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

76. 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)

77. i) Solutions : : linearly independent The general solution:

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

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

80. 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:

81. iii) Let The general sol.:

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

83. Maclaurin expansions:

84. 。 Eq. (2.5),

85. Find any two independent solutions Take
The general sol.:

86. 2.5. Euler’s Equation , A , B : constants -----(2.7)
Transform (2.7) to a constant coefficient equation
by letting

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

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

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

90. 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

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

92. 。 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

93. ii) Nonhomogeneous solution Let

94. iii) The general solution:

95. (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

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

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

98. 。 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

99. Try Substituting into (A),

100. ○ 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

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

102. 2.6.3. Superposition Let be a solution of is a solution of (A)

103. 。 Example 2.25: The general solution: where homogeneous solutions

104. 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:

105. 。Example 3.2:
Consider

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

108. ○ 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

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

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

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

112. ＊ 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

113. ○ 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

114. 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)

115. Proof:
Let

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

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

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

120. 3.3. Shifting Theorems and Heaviside Function
3.3.1.The First Shifting Theorem
◎ Theorem 3.7:
○ Example 3.6: Given

121. ○ Example 3.8:

122. 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

123. 。 Laplace transform of heaviside function

124. 3.3.3 The Second Shifting Theorem
Proof:

125. ○ Example 3.11: Rewrite

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

129. 3.4. Convolution

130. ◎ Theorem 3.9: Convolution theorem Proof:

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

132. ○ Example 3.19:

133. 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

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

135. Proof:

136. by Hospital’s rule
○ Example 3.20:

137. 3.6 Laplace Transform Solution of Systems
○ Example 3.22
Laplace transform
Solve for

138. Partial fractions decomposition Inverse Laplace transform

139. 3.7. Differential Equations with Polynomial Coefficient
◎ Theorem 3.13:
Proof:
○ Corollary 3.1:

140. ○ Example 3.25: Laplace transform

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

142. Inverse Laplace transform

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

144. ◎ Theorem 3.14: PC on [0, k],

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

146. Finding an integrating factor, Multiply (B) by ,

147. In order to have

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

149. ○Shifting Theorems: ○ Convolution: Convolution Theorem: ○

150. ○