Chapter 10 Systems of Nonlinear FirstOrder Differential Equations

Chapter 10 Systems of Nonlinear First-Order Differential Equations Dennis G. Zill, Differential Equations with

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

FIGURE 10. 1. 1 Vector field of a fluid flow in Example 3 Dennis

FIGURE 10. 1. 2 Curve in (a) is called an arc Dennis G. Zill,

FIGURE 10. 1. 3 Periodic solution or cycle Dennis G. Zill, Differential Equations with

FIGURE 10. 1. 4 Solution curves in Example 5 Dennis G. Zill, Differential Equations

FIGURE 10. 1. 5 Solution curve in Example 6 Dennis G. Zill, Differential Equations

FIGURE 10. 1. 6 Solution curves in Example 7 Dennis G. Zill, Differential Equations

FIGURE 10. 2. 1 Critical points Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 2 Phase portraits of linear system in Example 1 for various

FIGURE 10. 2. 3 Stable node Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 4 Unstable node Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 5 Saddle point Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 6 Saddle point Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 7 Stable node Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 8 Degenerate stable nodes Dennis G. Zill, Differential Equations with Boundary

FIGURE 10. 2. 9 Center Dennis G. Zill, Differential Equations with Boundary Value Problems,

FIGURE 10. 2. 10 Spiral points Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 11 Critical points in Example 3 Dennis G. Zill, Differential Equations

FIGURE 10. 2. 12 Geometric summary of Cases I, II, and III Dennis G.

FIGURE 10. 3. 1 Bead sliding on graph of z = f(x) Dennis G.

FIGURE 10. 3. 2 Critical points Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 3. 3 Asymptotically stable critical point in Example 1 Dennis G. Zill,

FIGURE 10. 3. 4 Unstable critical point in Example 2 Dennis G. Zill, Differential

FIGURE 10. 3. 5 In Example 3, /4 is asymptotically stable and 5 /4

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

FIGURE 10. 3. 7 Geometric summary of some conclusions (see (i)) and some unanswered

FIGURE 10. 3. 8 Phase portrait of nonlinear system in Example 8 Dennis G.

FIGURE 10. 3. 9 Phase portrait of nonlinear system in Example 9 Dennis G.

FIGURE 10. 4. 1 (0, 0) is stable and ( , 0) is unstable

FIGURE 10. 4. 2 Phase portrait of pendulum; wavy curves indicate that the pendulum

FIGURE 10. 4. 3 Some forces acting on sliding bead Dennis G. Zill, Differential

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

FIGURE 10. 4. 5 β = 0. 01 in Example 2 Dennis G. Zill,

FIGURE 10. 4. 6 β = 0 in Example 2 Dennis G. Zill, Differential

FIGURE 10. 4. 7 Solutions near (0, 0) Dennis G. Zill, Differential Equations with

FIGURE 10. 4. 8 Graphs of F and G help to establish properties (1)–(3)

FIGURE 10. 4. 9 Periodic solution of the Lotka-Volterra model Dennis G. Zill, Differential

FIGURE 10. 4. 10 Phase portrait of the Lotka-Volterra model in Example 3 Dennis

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

FIGURE 10. R. 1 Rotating pendulum in Problem 20 Dennis G. Zill, Differential Equations

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Chapter 10 Systems of Nonlinear First-Order Differential Equations Dennis G. Zill, Differential Equations with

Chapter 10 Systems of Nonlinear First-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 10. 1. 1 Vector field of a fluid flow in Example 3 Dennis

FIGURE 10. 1. 1 Vector field of a fluid flow in Example 3 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 3

FIGURE 10. 1. 2 Curve in (a) is called an arc Dennis G. Zill,

FIGURE 10. 1. 2 Curve in (a) is called an arc Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 4

FIGURE 10. 1. 3 Periodic solution or cycle Dennis G. Zill, Differential Equations with

FIGURE 10. 1. 3 Periodic solution or cycle Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 5

FIGURE 10. 1. 4 Solution curves in Example 5 Dennis G. Zill, Differential Equations

FIGURE 10. 1. 4 Solution curves in Example 5 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 6

FIGURE 10. 1. 5 Solution curve in Example 6 Dennis G. Zill, Differential Equations

FIGURE 10. 1. 5 Solution curve in Example 6 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 7

FIGURE 10. 1. 6 Solution curves in Example 7 Dennis G. Zill, Differential Equations

FIGURE 10. 1. 6 Solution curves in Example 7 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 8

FIGURE 10. 2. 1 Critical points Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 1 Critical points Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 9

FIGURE 10. 2. 2 Phase portraits of linear system in Example 1 for various

FIGURE 10. 2. 2 Phase portraits of linear system in Example 1 for various values of c Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 10

FIGURE 10. 2. 3 Stable node Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 3 Stable node Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 11

FIGURE 10. 2. 4 Unstable node Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 4 Unstable node Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 12

FIGURE 10. 2. 5 Saddle point Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 5 Saddle point Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 13

FIGURE 10. 2. 6 Saddle point Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 6 Saddle point Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 14

FIGURE 10. 2. 7 Stable node Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 7 Stable node Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 15

FIGURE 10. 2. 8 Degenerate stable nodes Dennis G. Zill, Differential Equations with Boundary

FIGURE 10. 2. 8 Degenerate stable nodes Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 16

FIGURE 10. 2. 9 Center Dennis G. Zill, Differential Equations with Boundary Value Problems,

FIGURE 10. 2. 9 Center Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 17

FIGURE 10. 2. 10 Spiral points Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 2. 10 Spiral points Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 18

FIGURE 10. 2. 11 Critical points in Example 3 Dennis G. Zill, Differential Equations

FIGURE 10. 2. 11 Critical points in Example 3 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 19

FIGURE 10. 2. 12 Geometric summary of Cases I, II, and III Dennis G.

FIGURE 10. 2. 12 Geometric summary of Cases I, II, and III Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 20

FIGURE 10. 3. 1 Bead sliding on graph of z = f(x) Dennis G.

FIGURE 10. 3. 1 Bead sliding on graph of z = f(x) Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 21

FIGURE 10. 3. 2 Critical points Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 10. 3. 2 Critical points Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 22

FIGURE 10. 3. 3 Asymptotically stable critical point in Example 1 Dennis G. Zill,

FIGURE 10. 3. 3 Asymptotically stable critical point in Example 1 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 23

FIGURE 10. 3. 4 Unstable critical point in Example 2 Dennis G. Zill, Differential

FIGURE 10. 3. 4 Unstable critical point in Example 2 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 24

FIGURE 10. 3. 5 In Example 3, /4 is asymptotically stable and 5 /4

FIGURE 10. 3. 5 In Example 3, /4 is asymptotically stable and 5 /4 is unstable Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 25

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

Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 26

FIGURE 10. 3. 7 Geometric summary of some conclusions (see (i)) and some unanswered

FIGURE 10. 3. 7 Geometric summary of some conclusions (see (i)) and some unanswered questions (see (ii) and (iii)) about nonlinear autonomous systems Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 27

FIGURE 10. 3. 8 Phase portrait of nonlinear system in Example 8 Dennis G.

FIGURE 10. 3. 8 Phase portrait of nonlinear system in Example 8 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 28

FIGURE 10. 3. 9 Phase portrait of nonlinear system in Example 9 Dennis G.

FIGURE 10. 3. 9 Phase portrait of nonlinear system in Example 9 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 29

FIGURE 10. 4. 1 (0, 0) is stable and ( , 0) is unstable

FIGURE 10. 4. 1 (0, 0) is stable and ( , 0) is unstable Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 30

FIGURE 10. 4. 2 Phase portrait of pendulum; wavy curves indicate that the pendulum

FIGURE 10. 4. 2 Phase portrait of pendulum; wavy curves indicate that the pendulum is whirling about its pivot Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 31

FIGURE 10. 4. 3 Some forces acting on sliding bead Dennis G. Zill, Differential

FIGURE 10. 4. 3 Some forces acting on sliding bead Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 32

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

Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 33

FIGURE 10. 4. 5 β = 0. 01 in Example 2 Dennis G. Zill,

FIGURE 10. 4. 5 β = 0. 01 in Example 2 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 34

FIGURE 10. 4. 6 β = 0 in Example 2 Dennis G. Zill, Differential

FIGURE 10. 4. 6 β = 0 in Example 2 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 35

FIGURE 10. 4. 7 Solutions near (0, 0) Dennis G. Zill, Differential Equations with

FIGURE 10. 4. 7 Solutions near (0, 0) Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 36

FIGURE 10. 4. 8 Graphs of F and G help to establish properties (1)–(3)

FIGURE 10. 4. 8 Graphs of F and G help to establish properties (1)–(3) Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 37

FIGURE 10. 4. 9 Periodic solution of the Lotka-Volterra model Dennis G. Zill, Differential

FIGURE 10. 4. 9 Periodic solution of the Lotka-Volterra model Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 38

FIGURE 10. 4. 10 Phase portrait of the Lotka-Volterra model in Example 3 Dennis

FIGURE 10. 4. 10 Phase portrait of the Lotka-Volterra model in Example 3 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 39

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

Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 40

FIGURE 10. R. 1 Rotating pendulum in Problem 20 Dennis G. Zill, Differential Equations

FIGURE 10. R. 1 Rotating pendulum in Problem 20 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 41