Chapter 5 Modeling with Higher Order Differential Equations

1
Chapter 5 Modeling with Higher Order
Differential Equations
Chapter 4 ????
???,????????? linear DE ???
???????????????? linear DE with constant
coefficients
2
5-1 Linear Models Initial Value Problem
??x, ?? ???
v ??, ?v ???
3
5-1-1 5-1-3 Spring / Mass Systems
4
Solution
5
?? F
???
6
???????
7
???????
(1) ???
??? input ? deriving function ? forcing function

??? output ? response
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(2) ???
??? 2nd order linear DE with constant coefficients
auxiliary function
? ?,?? overdamped
? ?,?? critical damped
? ?,?? underdamped
? , a1 0 ?,?? undamped
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(3)
?? a1 ????????? ? a2 , a1 , a0
?????, a1/ a2 ????,???????
When
When
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5-1-4 RLC circuit
inductance ???
capacitor ???
resistor ???
using
?????
11
auxiliary function
roots
Case 1 R2 ? 4L/C gt 0 (m1 ? m2, m1,
m2 are real)
(overdamped)
Complementary function
??? R, L, C ??????,
?????? ?? m1, m2 ????
when t ? ?
12
Particular solution (1) E(t) ??????guess
????? (2) E(t) ? variation of parameters
?????????(????)
13
Specially, when E(t) E0 where E0 is some
constant
(m1m2 1/LC)
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Case 2 R2 ? 4L/C 0 (m1 m2
?R/2L)
(critically damped)
when t ? ?
Particular solution
When E(t) E0 ,
15
(underdamped)
Case 3 R2 ? 4L/C lt 0
when t ? ?
Particular solution
General solution
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When E(t) E0 where E0 is some constant
When R 0 , then ? 0
When R 0 , E(t) E0
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? DE ?????? RLC ?????
(1) R2 lt 4L/C ????
(2) R ??,???????
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??
E(t) 1, L 0.25, C 0.01
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R 100
R 25
R 10
20
R 5
R 1.5
R 0.2
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5-1-5 Express the Solutions by Other Forms
(1) Express the Solution by the Form of Amplitude
and Phase
? ?,solution ?
Solution ????
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damped amplitude
damped frequency
phase angle
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(2) Express the Solution by Hyperbolic Functions
? a1 0 ? a2 gt 0, a0 lt 0 (? a0 gt 0, a2 lt 0)
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5-1-6 ????????
(1) ??????? DE ?,?????? (??????) (2) ?? page 274
??????? (3) ?? linear DE with constant
coefficients ??,?????? (see pages 287 and 289)
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5-2 Linear Models Boundary-Value Problem
(??????)
Section 5-2 ???,? Section 5-1 ?? (?? Linear
DE) ??? initial value problems ?? boundary value
problems
?? ? IVP ?? boundary value problems,
? solution ??????
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Section 5-1 ???
(1) ?????? (2) ???? (subsection
5-1-15-1-3) (3) RLC Circuit (subsection 5-1-4)
Section 5-2 ???
(1) ??? (a) ?? (subsection 5-2-1)
(b) ???? (subsection 5-2-2) (2) ??
(subsection 5-2-3)
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4-9 Solving Systems of Linear Equations by
Elimination
4-9-1 ??????????
??? 2 ??? dependent variables ???
??Section 3-3 ???? ?? ???
????? linear and constant coefficients
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4-9-2 ??
(Step 1) ?? ?? D n (Step 2)
??????(?????)??? ??? dependent
variable ???? DE ?? (Step 3) ??? Sections 4-3,
4-4 ???, ??? dependent variables
?? (Step 4) ????,?? unknowns ?????
(??????,?????? page 298)
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4-9-3 ??
Figure 3.3.4 ??? (See Pages 96, 97)
? L1 L2 1, R1 4, R2 6, E(t) 10
??
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.. (? 1)
Step 1
.. (? 2)
(D 4) ? (? 1) - 4 ? (? 2)
Step 2-1
Note
auxiliary function
Step 3-1
roots
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4 ? (? 1) - (D10) ? (? 2)
Step 2-2
auxiliary function
Step 3-2
roots
complementary function for i3,c(t)
Particular solution i3, p(t) A
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Step 4
??(? 1)
?
?? i2(t) ? i3(t) ??
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???????????
?
??(? 2)
?? c1 ?c2 ?????,?????
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? ? ??????
Step 2-2 ? i2(t) ????? ??? i2(t)
?? (?1)
????????????????? (?????????? dependent variable
i3(t) ???????)
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Example on text page 184
Example 1 (text page 185)
Example 3 (text page 186)
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Example 2 (text page 185)
(?1) (?2)
Step 1
Step 2-1 (?2) ?D - (?1)
complementary function
Step 3-1
particular solution
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Step 2-2 (?1) ?(D1) - (?2) ? (D-4)
Step 3-2
complementary function
particular solution
??,????
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Step 4 (???2) (???2??1??)
?
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4-9-4 ?? Dependent Variables
Exercise 19
(?1)
(?2 )
(?3 )
Steps 2, 3???????? x, y, z ? DE
(?4 )
(?2) ? D (?3)
(?4) ? 6 (?1) ? (1-D2)
m -1, -2, 3
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(?4) ? D - (?1) ? (1D)
(?3) ? 6 (?1)
(?5 )
(?1) ? D - (?2) ? 6
(?6 )
(?5) ? (D2 -6) (?6) ? (D6)
Step 4? c4, c5, c6, c7, c8, c9 ? c1, c2, c3 ??
?
?? (?1)
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?
?? (?2)
??y ? z ???????
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Higher order ??
(1) ??? N ? dependent variables, ?????? N ?
DE ????? solutions (2) ???? DE ? orders ??? k1,
k2, k3, .., kN ????????? order ? k1 k2
k3 .. kN ? DE (3) ????? N -1 ??,?? unknowns
?????
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A circuit that can be modeled by a 2nd order
polynomial.
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4-9-5 ????????
(1) Section 4.9 ??????? constant coefficients
??? (2) ??? dependent variable ??,
homogeneous ???????????? (3) ????,?????
(??????????????????) (4) ?? (5) ??? Step 4 ??
unknowns ????? (6) ????????????
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4-10 Nonlinear Differential Equations
Method 1 Reduction of Order
Method 2 Taylor Series
Method 3 Numerical Approach
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4-10-1 Method 1 Reduction of Order
???? 1st order DE ?? 1st order DE
?????
(??????? Section 4-2 ??,????? linear,
???????????)
??The DE should have the form of
Case 1, page 313
Case 2, page 315
or
(Without the term x)
(Without the term y)
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Case 1 The 2nd order DE has the form of
(Without the term y)
u
?? (Step 1) Set ??DE ??
(? u ??,? 1st order DE)
(Step 2) ? u ??? (? Section 2 ???)
(Step 3) ? u ???,??? y
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Example 1 (text page 189)
(Step 1)
?? u ????????
(Step 2)
(Step 3)
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Case 2 The 2nd order DE has the form of
(Without the term x)
u
??(Step 1) Set

(Chain rule) ??DE
?? (?
u ??,?1st order DE, independent variable ? y)

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(Step 2) ? u ??? (? Section 2 ???)
????, u ? y ??? (Step 3)
? separable variable ?????????
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Example 2 (text page 190)
(Step 1) Set
(Step 2)
(Step 3)
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4-10-2 Method 2 Taylor Series
???????
Step 1 ??
Step 2 ?? Taylor series
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Example 3 (text page 190)

?? Taylor series
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??(1) y(x) ? x0 ?????? analytic,
(x x0 ?? singular point)
(2)
?? nth order DE ?,y(x0), y'(x0), y''(x0), ..
y(n?1)(x0) ????????
(3) ??????? x0 ??????
??(1) Taylor series ??????? (2) x
?x0 ????
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x0 0
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4-10-3 Method 3 Numerical Method
subject to
??
??Section 2-6 ? Eulers Method
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Recursive ???
Initial
n 0
n n 1
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???????

???
subject to
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Recursive ???
n n 1
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?????
(1) ?
???? (?? singular point) ??
???????,??????
??????????? (2) ??? k ??????initial conditions
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4-10-4 ????????
(1) Section 4.10 ?????????????
??????????? (2) Section 4.1 ????????? (??
exercises 4.10 ?? 1, 2 ?) (3) Method 1
??????,????? (4) Method 1 ???? u ? dy/dx ??
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5-3 Nonlinear Models
???????? (text pages 222, 223) ????? (text pages
223, 224, 225) ?????? (text pages 225, 226) ?????
(text page 226) ?????? (text pages 227, 228)
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5-3-1 ?????
F ma
F ??? y ??? (??????)
????
M ?????
m ?????
? ??
???
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5-3-2 ??????
m ??, v ??, mv ??
20 N
m ??? x ??? (??????), m kx(t)
In Example 4, chain weight 1 N/m
?? (weight) x(t)
?? (mass) x(t)/9.8
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F0 ??, k ???????, x(t) ?? (???)
????????, g 9.8 metres per s2
?????, g 32 feet per s2
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Example 4 (text page 227)
k 1/9.8 g 9.8 F0 20
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(1) Cauchy-Euler equation ???? (2) ?????????
Numerical Method ??
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5-3-3 ????????
(1) ??????,??? DE ?????? ???????????? DE
???? (2) ??? (?????)??? (3) tan? ?? (4)
??????????????????
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Reviews for Higher Order DE (A) Linear DE
Complementary Function 3 ??? (1) Reduction of
Order (Section 4-2) ???? (2)
Auxiliary Function (Section 4-3)
????
4 Cases (See pages 182, 183)
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(3) Cauchy-Euler Equation (Section 4-7)
????
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(B) Linear DE Particular solution 3 ???
(1) Guess (Section 4-4) (???? page 194 ??)
?? yp should be a linear combination of g(x),
g'(x), g'' (x), g'''(x),
g(4)(x), g(5)(x), .
????
????,? x ? lnx
(2) Annihilator (Section 4-5)
???? DE ? Ly(x) g(x)
Annihilator L1 g(x) 0
Particular solution ? L1Ly(x) 0 ??
(??? Ly(x) 0 ???????)
????
Annihilator ?????? Pages 213-215
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(3) Variation of parameters (Section 4-6)
Wk replace the kth column of W by
????
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(4) For Cauchy-Euler Equation (Section 4-7)
???????
(1) ??
? complementary function
?? Variation of parameters ? particular solution
(2) Use the method on pages 261, 262
Set x et, t ln x
then
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(C) Combination of Linear DEs (Section 4-9)
?? Step 1 ? ?? Dn Steps 2, 3
??????,??????? dependent variable
? DE,???? dependent variable ??? Step 4
????,?? dependent variable ??? ck ???
??
????
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(D) Nonlinear DE ?3??? (Section 4-10)
(1) Reduction of Order (1-1)
Set (1-2)
Set
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(2) Taylor Series ???? (3) Numerical
Method ????
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Exercises for practicing
Section 4-9 5, 8, 10, 14, 17, 18, 20, 22,
23 Section 4-10 1, 4, 5, 8, 10, 12, 15, 16,
19, 21, 22, 23 Review 4 43, 44, 48,
50 Section 5-1 1, 11, 29, 43, 44, 49, 52, 56,
60 Section 5-3 14, 15, 16
Review 5 12, 21, 22
???? 2012???????,2016, 2017?????