Chapter 5 Modeling with Higher Order Differential Equations

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Chapter 5 Modeling with Higher -Order Differential Equations Dennis G. Zill, Differential Equations with

Chapter 5 Modeling with Higher -Order Differential Equations Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018

Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 2

Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 2

FIGURE 5. 1. 1 Spring/mass system Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 5. 1. 1 Spring/mass system Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 3

FIGURE 5. 1. 2 Direction below the equilibrium position is positive Dennis G. Zill,

FIGURE 5. 1. 2 Direction below the equilibrium position is positive Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 4

FIGURE 5. 1. 3 A relationship between c 1 > 0, c 2 >

FIGURE 5. 1. 3 A relationship between c 1 > 0, c 2 > 0 and phase angle Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 5

FIGURE 5. 1. 4 Simple harmonic motion Dennis G. Zill, Differential Equations with Boundary

FIGURE 5. 1. 4 Simple harmonic motion Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 6

FIGURE 5. 1. 5 Parallel springs Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 5. 1. 5 Parallel springs Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 7

FIGURE 5. 1. 6 Springs in series Dennis G. Zill, Differential Equations with Boundary

FIGURE 5. 1. 6 Springs in series Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 8

FIGURE 5. 1. 7 Damping devices Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 5. 1. 7 Damping devices Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 9

FIGURE 5. 1. 8 Motion of an overdamped system Dennis G. Zill, Differential Equations

FIGURE 5. 1. 8 Motion of an overdamped system Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 10

FIGURE 5. 1. 9 Motion of a critically damped system Dennis G. Zill, Differential

FIGURE 5. 1. 9 Motion of a critically damped system Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 11

FIGURE 5. 1. 10 Motion of an underdamped system Dennis G. Zill, Differential Equations

FIGURE 5. 1. 10 Motion of an underdamped system Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 12

FIGURE 5. 1. 11 Overdamped system in Example 3 Dennis G. Zill, Differential Equations

FIGURE 5. 1. 11 Overdamped system in Example 3 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 13

FIGURE 5. 1. 12 Critically damped system in Example 4 Dennis G. Zill, Differential

FIGURE 5. 1. 12 Critically damped system in Example 4 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 14

FIGURE 5. 1. 13 Oscillatory vertical motion of the support Dennis G. Zill, Differential

FIGURE 5. 1. 13 Oscillatory vertical motion of the support Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 15

FIGURE 5. 1. 14 Graph of solution in (28) of Example 6 Dennis G.

FIGURE 5. 1. 14 Graph of solution in (28) of Example 6 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 16

FIGURE 5. 1. 15 Graph of solution in Example 7 for various initial velocities

FIGURE 5. 1. 15 Graph of solution in Example 7 for various initial velocities x 1 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 17

FIGURE 5. 1. 16 Pure resonance Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 5. 1. 16 Pure resonance Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 18

FIGURE 5. 1. 17 LRC-series circuit Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 5. 1. 17 LRC-series circuit Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 19

FIGURE 5. 1. 18 Graph for Problem 21 Dennis G. Zill, Differential Equations with

FIGURE 5. 1. 18 Graph for Problem 21 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 20

FIGURE 5. 1. 19 Graph for Problem 22 Dennis G. Zill, Differential Equations with

FIGURE 5. 1. 19 Graph for Problem 22 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 21

FIGURE 5. 1. 20 Graph for Problem 23 Dennis G. Zill, Differential Equations with

FIGURE 5. 1. 20 Graph for Problem 23 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 22

FIGURE 5. 1. 21 Graph for Problem 24 Dennis G. Zill, Differential Equations with

FIGURE 5. 1. 21 Graph for Problem 24 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 23

FIGURE 5. 1. 22 Oscillating support in Problem 39 Dennis G. Zill, Differential Equations

FIGURE 5. 1. 22 Oscillating support in Problem 39 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 24

FIGURE 5. 1. 23 Beats phenomenon in Problem 45 Dennis G. Zill, Differential Equations

FIGURE 5. 1. 23 Beats phenomenon in Problem 45 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 25

FIGURE 5. 2. 1 Deflection of a homogeneous beam Dennis G. Zill, Differential Equations

FIGURE 5. 2. 1 Deflection of a homogeneous beam Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 26

FIGURE 5. 2. 2 Beams with various end conditions Dennis G. Zill, Differential Equations

FIGURE 5. 2. 2 Beams with various end conditions Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 27

TABLE 5. 2. 1 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9

TABLE 5. 2. 1 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 28

FIGURE 5. 2. 3 Deflection curve for BVP in Example 1 Dennis G. Zill,

FIGURE 5. 2. 3 Deflection curve for BVP in Example 1 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 29

FIGURE 5. 2. 4 Graphs of eigenfunctions yn = sin(n x/L), for n =

FIGURE 5. 2. 4 Graphs of eigenfunctions yn = sin(n x/L), for n = 1, 2, 3, 4, 5 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 30

FIGURE 5. 2. 5 Elastic column buckling under a compressive force Dennis G. Zill,

FIGURE 5. 2. 5 Elastic column buckling under a compressive force Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 31

FIGURE 5. 2. 6 Deflection curves corresponding to compressive forces P 1, P 2,

FIGURE 5. 2. 6 Deflection curves corresponding to compressive forces P 1, P 2, P 3 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 32

FIGURE 5. 2. 7 Rotating string and forces acting on it Dennis G. Zill,

FIGURE 5. 2. 7 Rotating string and forces acting on it Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 33

FIGURE 5. 2. 8 Deflection of cantilever beam in Problem 7 Dennis G. Zill,

FIGURE 5. 2. 8 Deflection of cantilever beam in Problem 7 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 34

FIGURE 5. 2. 9 Deflection of vertical column in Problem 24 Dennis G. Zill,

FIGURE 5. 2. 9 Deflection of vertical column in Problem 24 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 35

FIGURE 5. 2. 10 Concentric spheres in Problem 29 Dennis G. Zill, Differential Equations

FIGURE 5. 2. 10 Concentric spheres in Problem 29 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 36

FIGURE 5. 2. 11 Circular ring in Problem 30 Dennis G. Zill, Differential Equations

FIGURE 5. 2. 11 Circular ring in Problem 30 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 37

FIGURE 5. 2. 12 Propeller shaft of the battleship USS Missouri Dennis G. Zill,

FIGURE 5. 2. 12 Propeller shaft of the battleship USS Missouri Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 38

FIGURE 5. 3. 1 Hard and soft springs Dennis G. Zill, Differential Equations with

FIGURE 5. 3. 1 Hard and soft springs Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 39

FIGURE 5. 3. 2 Numerical solution curves Dennis G. Zill, Differential Equations with Boundary

FIGURE 5. 3. 2 Numerical solution curves Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 40

FIGURE 5. 3. 3 Simple pendulum Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 5. 3. 3 Simple pendulum Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 41

FIGURE 5. 3. 4 In Example 2, oscillating pendulum in (b); whirling pendulum in

FIGURE 5. 3. 4 In Example 2, oscillating pendulum in (b); whirling pendulum in (c) Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 42

FIGURE 5. 3. 5 Shape of hanging telephone wires is a catenary Dennis G.

FIGURE 5. 3. 5 Shape of hanging telephone wires is a catenary Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 43

FIGURE 5. 3. 6 Distance to rocket is large compared to R. Dennis G.

FIGURE 5. 3. 6 Distance to rocket is large compared to R. Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 44

FIGURE 5. 3. 7 Chain pulled upward by a constant force in Example 4

FIGURE 5. 3. 7 Chain pulled upward by a constant force in Example 4 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 45

FIGURE 5. 3. 8 Graph of (21) in Example 4 Dennis G. Zill, Differential

FIGURE 5. 3. 8 Graph of (21) in Example 4 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 46

FIGURE 5. 3. 9 Pursuit curve in Problem 17 Dennis G. Zill, Differential Equations

FIGURE 5. 3. 9 Pursuit curve in Problem 17 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 47

FIGURE 5. 3. 10 Pursuit curve in Problem 18 Dennis G. Zill, Differential Equations

FIGURE 5. 3. 10 Pursuit curve in Problem 18 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 48

FIGURE 5. 3. 11 Ballistic pendulum in Problem 19 Dennis G. Zill, Differential Equations

FIGURE 5. 3. 11 Ballistic pendulum in Problem 19 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 49

FIGURE 5. 3. 12 Airplane drop in Problem 20 Dennis G. Zill, Differential Equations

FIGURE 5. 3. 12 Airplane drop in Problem 20 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 50

FIGURE 5. R. 1 Rotating rod in Problem 24 Dennis G. Zill, Differential Equations

FIGURE 5. R. 1 Rotating rod in Problem 24 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 51

FIGURE 5. R. 2 Sliding spring/mass system in Problem 25 Dennis G. Zill, Differential

FIGURE 5. R. 2 Sliding spring/mass system in Problem 25 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 52

FIGURE 5. R. 3 Double spring system in Problem 26 Dennis G. Zill, Differential

FIGURE 5. R. 3 Double spring system in Problem 26 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 53

FIGURE 5. R. 4 Spring pendulum in Problem 30 Dennis G. Zill, Differential Equations

FIGURE 5. R. 4 Spring pendulum in Problem 30 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 54

FIGURE 5. R. 5 Pendulum in Problem 31 Dennis G. Zill, Differential Equations with

FIGURE 5. R. 5 Pendulum in Problem 31 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 55

FIGURE 5. R. 6 Collapse of the Tacoma Narrows Suspension Bridge Dennis G. Zill,

FIGURE 5. R. 6 Collapse of the Tacoma Narrows Suspension Bridge Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 56