1. Advanced Engineering Mathematics 6th Edition, Concise Edition
by Peter V. O’Neil

2. 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

3. 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

4. (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

5. 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

6. 1.1.1. General and Particular Solutions○
General solution:
arbitrary constant
Substitute into (A)
Particular solutions:
k = 1, ; k = 2 ,
k = ,

7. 1.1.2. Implicitly Defined Solutions
○ Explicit function:
Implicit function:
e.g., , ○
(Explicit solution)
(Implicit solution)

8. 1.1.3. Integral Curves ○ Example 1.1: General solution:
-- Help to comprehend the behavior of solution
○ Example 1.1:
General solution:

9. 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: ,

10. -- 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

11. -- 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 11

12. 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).

13. 1.2. Separable Equations

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

15. 1.3. Linear Differential Equations -------- (A)
(1) Find integrating factor:
(2) Multiply (A) by
(3)

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

17. 1.4. Exact Differential Equations
can be written as
(A)
If , s.t.
and
(implicitly define the solution)
: potential function

18. ○ Example 1.17: Let From

19. ＊ We can start with , then as well
From Solution:
＊ We can start with , then as well 19

20. 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

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

22. Let
: any point

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

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

25. Consider Let Try (B) Integrate

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

27. ○ Example 1.22:
Let
Find integrating factor by

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

29. (iii) Try (B) Divide by

30. : independent
Multiply (A) by
Let
Form

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

32. 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)

33. ○ Example 1.25: Let (A)

34. 1.6.2. Bernoulli Equation linear separable
can be transformed into linear by

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

36. 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

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

38. 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.

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

40. ○ 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

41. 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 41

42. ○ 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

43. 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

44. 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,

45. must be independent of x.
The test is then that
must be independent of x. 45

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

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

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

49. The graph of Given the initial condition

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

51. ○ 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:

52. ※ 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

53. ○ 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

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

55. ○ 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.

56. (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

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

58. 。 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

59. ○ 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

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

61. 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

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

63. 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

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

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

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

67. 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)

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

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

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

71. 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:

72. iii) Let The general sol.:

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

74. Maclaurin expansions:

75. 。 Eq. (2.5),

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

77. 2.5. Euler’s Equation , A , B : constants -----(2.7)
Transform (2.7) to a constant coefficient equation
by letting

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

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

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

81. 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

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

83. 。 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

84. ii) Nonhomogeneous solution Let

85. iii) The general solution:

86. (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

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

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

89. 。 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

90. Try Substituting into (A),

91. ○ 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

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

93. 2.6.3. Superposition Let be a solution of is a solution of (A)

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

95. 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:

96. 。Example 3.2:
Consider

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

99. ○ 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

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

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

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

103. ＊ 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

104. ○ 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

105. 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)

106. Proof:
Let

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

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

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

111. 3.3. Shifting Theorems and Heaviside Function
3.3.1.The First Shifting Theorem
◎ Theorem 3.7:
○ Example 3.6: Given

112. ○ Example 3.8:

113. 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

114. 。 Laplace transform of heaviside function

115. 3.3.3 The Second Shifting Theorem
Proof:

116. ○ Example 3.11: Rewrite

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

120. 3.4. Convolution

121. ◎ Theorem 3.9: Convolution theorem Proof:

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

123. ○ Example 3.19:

124. 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

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

126. Proof:

127. by Hospital’s rule
○ Example 3.20:

128. 3.6 Laplace Transform Solution of Systems
○ Example 3.22
Laplace transform
Solve for

129. Partial fractions decomposition Inverse Laplace transform

130. 3.7. Differential Equations with Polynomial Coefficient
◎ Theorem 3.13:
Proof:
○ Corollary 3.1:

131. ○ Example 3.25: Laplace transform

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

133. Inverse Laplace transform

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

135. ◎ Theorem 3.14: PC on [0, k],

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

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

138. In order to have

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

140. ○Shifting Theorems: ○ Convolution: Convolution Theorem: ○

141. ○