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MATH2010 Analysis of Ordinary Differential Equations

WORKBOOK Semester 1, 2019

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c School of Mathematics and Physics, The University of Queensland, Brisbane QLD 4072, Australia. 

2

CONTENTS

Contents 1 Differential Equations

5

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.1.1

Electrical Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.1.2

Systems of ODE’s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.1.3

Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.1.4

Texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.2 Introduction to Systems of ODE’s and Classification of ODE’s. . . . . . . . 10

1.3

1.4

1.2.1

Introduction to Systems of ODE’s . . . . . . . . . . . . . . . . . . . 10

1.2.2

Classifying ODE’s: Linear, Order, Homogeneous. . . . . . . . . . . 13

1.2.3

The Superposition Principle . . . . . . . . . . . . . . . . . . . . . . 16

Solving systems of two coupled 1st order ODE’s. . . . . . . . . . . . . . . 16 1.3.1

The system in matrix form. . . . . . . . . . . . . . . . . . . . . . . 16

1.3.2

The Homogeneous case with constant coefficients. . . . . . . . . . . 17

Theory and Theorems for first order systems. . . . . . . . . . . . . . . . . 30

1.5 Homogeneous Constant Coefficient Linear 2-dimensional Systems and the Phase Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.5.1

The Phase Portrait

1.5.2

Phase Portraits for Real Eigenvalues and Direction Fields. . . . . . 36

1.5.3

Phase Portraits for Complex Eigenvalues

1.5.4

. . . . . . . . . . . . . . . . . . . . . . . . . . 34

. . . . . . . . . . . . . . 47

SUMMARY Of 6 Types of LINEAR PHASE PORTRAITS in 2D . 50

1.6 Critical Points and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.6.1

Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.6.2

Stability of Critical points. . . . . . . . . . . . . . . . . . . . . . . 53

1.7

Non homogeneous Linear systems . . . . . . . . . . . . . . . . . . . . . . . 57

1.8

Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.8.1

Solving for the Phase Curves . . . . . . . . . . . . . . . . . . . . . . 58

1.8.2

Critical Points for Nonlinear Systems. . . . . . . . . . . . . . . . . . 61

CONTENTS

1.9

3

1.8.3

Linearization of Nonlinear Systems. . . . . . . . . . . . . . . . . . . 62

1.8.4

Summary for Nonlinear Systems . . . . . . . . . . . . . . . . . . . 70

Diagonalization and 2D Phase Portraits . . . . . . . . . . . . . . . . . . . 71 1.9.1

Relevance to 2D Phase Portraits. . . . . . . . . . . . . . . . . . . . 71

2 Laplace Transforms 2.1

2.2

73

Finding the Laplace Transform of a function . . . . . . . . . . . . . . . . . 73 2.1.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.1.2

The Laplace Transform of simple functions. . . . . . . . . . . . . . 73

2.1.3

The Laplace Transform for Piecewise Continuous functions. . . . . . 77

2.1.4

The First Shifting Theorem. . . . . . . . . . . . . . . . . . . . . . . 79

2.1.5

Summary of Laplace Transforms . . . . . . . . . . . . . . . . . . . . 80

2.1.6

Inverting Laplace Transforms. . . . . . . . . . . . . . . . . . . . . . 81

2.1.7

The Gamma Function and L(ta), where a is not an integer. . . . . . 83

Laplace Transforms of Derivatives and Solving Simple Linear Constant Coefficient ODEs and Systems of ODE’s. . . . . . . . . . . . . . . . . . . . 84 2.2.1

The Laplace Transform of the differential of a function. . . . . . . . 84

2.2.2

Solving Linear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.2.3

Forcing Functions and Transfer Functions. . . . . . . . . . . . . . . 87

2.2.4

Solving Systems of ODE’s . . . . . . . . . . . . . . . . . . . . . . . 90

2.3 Finding the inverse Laplace Transform of complicated functions using Partial Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.3.1

Simple Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

2.3.2

Repeated simple factors . . . . . . . . . . . . . . . . . . . . . . . . 94

2.3.3

Irreducible Quadratic Factors . . . . . . . . . . . . . . . . . . . . . 95

2.3.4

Repeated irreducible factors and the Inverse Laplace Transform of dF (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 ds 2.4 The Second Shifting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.4.1

Using the Second Shifting Theorem . . . . . . . . . . . . . . . . . . 100

4

CONTENTS

2.4.2

Circuit Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2.5 The Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.5.1

Using the Convolution Theorem . . . . . . . . . . . . . . . . . . . . 111

2.5.2

Solving Linear ODE’s using Convolution. . . . . . . . . . . . . . . 113

2.5.3

The Laplace Transform of Periodic Functions . . . . . . . . . . . . 115

5

1 1.1

Differential Equations Introduction

Differential Equations come in two basic types; ODE’s and PDE’s. Ordinary Differential Equations & Partial Differential Equations .

(ODE’s) MATH2010

(PDE’s) MATH2011.

In ODE’s the unknown is a function of one independent variable, so that only ordinary differentials are involved. For example the equation for a mass spring system: m

d2 x dx = −kx − c 2 dt dt

is an equation for the unknown x as a function of the dependent variable t; x(t). In PDE’s the unknown is a function of 2 or more independent variables and there are partial differentials. For example the heat equation : m

∂ 2H ∂H =c ∂t ∂x2

is an equation for the unknown H as a function of the dependent variables x, t; H(x, t). MATH2010 has two parts; SYSTEMS OF ODE’s and LAPLACE TRANSFORMS SYSTEMS OF ODE’s often arise naturally. For instance to model the spread of an infectious disease the rate of change of the number of those infected depends on the number of of those who are susceptible. d Infected = f (Infected, Susceptible) dt d Susceptible = g(Infected, Susceptible) dt In an electrical circuit the rate of change of charge gives the current, but the rate of change of current itself depends on charge.

6

1. DIFFERENTIAL EQUATIONS

1.1.1

Electrical Circuit

R C L I(t) Q(t) E(t)

Kirchhoff ’s Law says that the voltage drop across the Inductor plus the voltage drop across the Resistor plus the voltage drop across the Capacitor equals the applied voltage. Q(t) dI(t) = E(t) + RI(t) + C dt But charge is related to current: L

So we could substitute this back to give one second order equation for Q(t).

Or we can write as a system of two coupled first order ODE’s.

1.1. INTRODUCTION

1.1.2

7

Systems of ODE’s

Take the Electrical circuit L

d2 Q(t) dQ(t) Q(t) = E(t) +R + 2 C dt dt

with R = 0, L = 1, C = 1 and E(t) = 0, then

If Q(0) = 0 this has solution

So the charge and current oscillate out of phase with each other. We can see from the equations that

Q2 + I 2 =

the solution curves are circles in (Q, I) space.

8

1. DIFFERENTIAL EQUATIONS

But the full 3-dimensional space (t, Q, I) = (t, c2 sin(t), c2 cos( t)) is really hard to work with.

So we have two options;

In the second option, (Q(t), I(t)) space, the curve representing the solutions is parametrized by time. Because the curve is a circle we can see that the behavior is cyclic and that when Q(t) is at a maximum I(t)) is 0 etc. But some information has been lost. For instance we don’t know how fast to move around the circle. The (Q(t), I(t)) space is called Phase Space.

1.1. INTRODUCTION

9

There are about 6 qualitatively different Linear systems in 2D Phase Space which we will look at and classify. Then we will begin to to ask what we can do with Nonlinear Systems. Search on the Internet for pplane and try entering your own linear system.

1.1.3

Laplace Transforms

In LAPLACE TRANSFORMS we will solve systems which are time dependent, such as the circuit above with a variable applied voltage. What makes Laplace Transforms so useful is that you can use them to solve equations with discontinuous terms, say a voltage that is suddenly switched on! L

d2 Q(t) dQ(t) Q(t) +R + = E(t) dt2 dt C

Suppose the voltage has been switched on and is then switched off. We could assume that it is switched off at say t = k

E(t) =



E0 0

0≤t 1 these 2

these curves for C 6= 0 have zero slope at the origin. So they approach the origin tangent to the y1 axis.

What if both eigenvalues are positive? Actually the situation isn’t much different, after all the two minuses cancelled when we solved for y2 as a function of y1 . Lets take an example though.      y1 1 3 y˙1 = 0 2 y˙2 y2 which has solution y = c1

Now the straight line solutions are



1 0



t

e + c2



3 1



e2t .

1.5. HOMOGENEOUS CONSTANT COEFFICIENT LINEAR 2-DIMENSIONAL SYSTEMS AND THE PHASE PLANE

39

In between the curves are tangent to the straight line corresponding to the eigenvector of the eigenvalue with least magnitude.

Any system with two positive real and different eigenvalues gives similar results and is called an -UNSTABLE (improper) NODE.

Direction Fields and Nullclines.



 y1 Considered as a vector equation: = f (y) with y = y2 f (y) evaluated at (y1 , y2 ) is the velocity vector at (y1 , y2 ). dy dt

The direction of the velocity vector gives the direction of the flow and the length of the velocity vector is the speed. The Direction Field is the field of these vectors: f (y). (’Graph Phase Plane’ in pplane gives you a direction field.)

The great thing about the direction field is that it can be calculated without ever solving the ODE. This is true for a nonlinear system as well as a linear one.

So for the system



y˙1 y˙2



=



1 3 0 2



y1 y2



⇒ f (y) =



y1 + 3y2 2y2



40

1. DIFFERENTIAL EQUATIONS

At say (y1 , y2 ) = (0, 2) or (y1 , y2 ) = (2, 0) or (y1 , y2 ) = (−2, 1)

The Slope of a Trajectory. Often the most useful aspect of the vector is its slope, f2 (y1 , y2 ) dy2 = given by . f1 (y1 , y2 ) dy1 The slope along y2 = 0 is or along y1 = 0 or along y2 = y1 1 or along y2 = − y1 3

Nullclines A Nullcline is a line or curve where the slope of the trajectory is 0 or ∞. (The package pplane will plot the nullclines for you. Go to Solution and then Show Nullclines in the phase plane window.) The Nullclines for the example above are

1.5. HOMOGENEOUS CONSTANT COEFFICIENT LINEAR 2-DIMENSIONAL SYSTEMS AND THE PHASE PLANE

41

What if the eigenvalues have opposite sign. Here again it is useful to be able to solve for y2 as a function of y1 , so I will take a simple example. where the eigenvectors lie on the axes.          1 y1 2 0 y˙1 0 2t e + c2 e−4t . ⇒ y = c1 = 0 1 0 −4 y2 y˙2 The straight line solutions lie on the axes: c2 = 0 ⇒ c1 = 0 ⇒

On the y1 axis we have growth, so the arrow goes away from the origin, while in the y2 axis we have decay, so the arrow comes into the origin. Also when we solve for y2 as a function of y1 ;

Once again this is typical of the case where the eigenvalues have opposite sign. Suppose the solution was     1 1 t e + c2 e−3t . y = c1 3 −1

A sketch of the trajectories gives the following.

Any system with one positive and one negative eigenvalue gives similar results and is called a -SADDLE.

42

1. DIFFERENTIAL EQUATIONS

SUMMARY for REAL and DISTINCT Eigenvalues

1. Straight line solutions. If the eigenvalues of A are real and distinct then the solution to y˙ = Ay is in the form y = c1 x(1) eλ1 t + c2 x(2) eλ2 t

AND there are two straight lines in the phase portrait associated with the solutions for c1 = 0 and c2 = 0.

2. Node or saddle

If the eigenvalues of A are both negative the origin is said to be a stable (improper) NODE.

If the eigenvalues of A are both positive the origin is said to be an unstable (improper) NODE.

1.5. HOMOGENEOUS CONSTANT COEFFICIENT LINEAR 2-DIMENSIONAL SYSTEMS AND THE PHASE PLANE

43

If the eigenvalues of A are of opposite sign the origin is said to be a SADDLE.

Have a look at mathsims the General Linear Model for another examples. 3. Direction Field Since f (y) evaluated at (y1 , y2 ) is the velocity vector at (y1 , y2 ), the slope of the curves in phase space is given by dy2 y˙2 = y˙1 dy1

(from the chain rule.)

=

a21 y1 + a22 y2 a11 y1 + a12y2

In particular the lines where the phase curves are horizontal and where they are vertical are called the Nullclines

4. The solutions to the equation dy2 a21 y1 + a22y2 = a11 y1 + a12y2 dy1

are the phase space curves.

44

1. DIFFERENTIAL EQUATIONS

If the eigenvalues are equal there may still be two linearly independent eigenvectors.

For instance if A =



1 0 0 1



So the General Solution is      0 1 + c2 et . =⇒ y = c1 0 1

This is called a PROPER NODE and it is unstable if λ > 0 and stable if λ < 0.

1.5. HOMOGENEOUS CONSTANT COEFFICIENT LINEAR 2-DIMENSIONAL SYSTEMS AND THE PHASE PLANE

45

Alternatively there may be only one eigenvector.   0 1 For instance for A = −1 2

Then there is only one straight line

y2 y1

= 1 in the phase plane.

To get some idea as to how the other trajectories come into the origin consider. dy2 −y1 + 2y2 = y2 dy1

which is the slope of the trajectory at the point(y1 , y2 )

Along y2 = 12 y1

but along y2 = 0

The trajectories bend around, almost spiraling, but not quite. This is called an DEGENERATE or INFLECTED NODE and it is unstable if λ > 0 and stable if λ < 0.

46

1.5.3

1. DIFFERENTIAL EQUATIONS

Phase Portraits for Complex Eigenvalues

If the eigenvalues are pure imaginary you can prove that the trajectories are ellipses. Take the following example      y1 0 1 y˙1 = −2 0 y2 y˙2 dy2 y˙2 = y˙1 dy1

Determining the direction of flow. Go back to the equation of motion for y1 ; and consider the sign of y˙1 on the upper part of the y2 axis.

Any system with pure imaginary eigenvalues is called a CENTER and has elliptical trajectories, but finding the actual elliptic orbit may be tricky (until you know about diagonalization).

1.5. HOMOGENEOUS CONSTANT COEFFICIENT LINEAR 2-DIMENSIONAL SYSTEMS AND THE PHASE PLANE

47

If the eigenvalues are complex with nonzero real part then the trajectories spiral in (if the real part is negative) or out (if the real part is positive). To prove this in general requires diagonalization, but it is easily seen in the following case. 

y˙1 y˙2



which has complex solutions

= 



2 −1 1 2

1 −i





e(2+i)t

y1 y2



and it’s complex conjugate.

So the real solutions are

For each solution y12 + y22 = e4t In polar coordinates (y1 = r cos θ, y2 = r sin θ )

So solutions spiral out. From a topological point of view there are two possibilities:

Determining the direction of flow. Go back to the equation of motion for y1 ; and consider the sign of y˙1 on the upper part of the y2 axis.

48

1. DIFFERENTIAL EQUATIONS

For a more exact picture of the flow you can use the slope of the trajectories as a guide. y1 + 2y2 dy2 = 2y1 − y2 dy1

Any system with complex eigenvalues is called a SPIRAL. If the real part of the eigenvalues is positive solutions spiral out from the origin and the critical point at the origin is UNSTABLE. If the real part of the eigenvalues is negative solutions spiral in to the origin and the critical point at the origin is STABLE.

1.5. HOMOGENEOUS CONSTANT COEFFICIENT LINEAR 2-DIMENSIONAL SYSTEMS AND THE PHASE PLANE

1.5.4

49

SUMMARY Of 6 Types of LINEAR PHASE PORTRAITS in 2D

The type depends on the eigenvalues and eigenvectors of A . (For more examples go to the mathsims website and click on the General Linear Model. To see the full picture with many sample trajectories, click on other initial conditions in the phase plane.) 1 SADDLE One positive eigenvalue and one negative eigenvalue.

2 IMPROPER NODE Two different positive eigenvalues UNSTABLE or different negative eigenvalues STABLE.

3 PROPER NODE Equal eigenvalues and two linearly independent eigenvectors.

50

1. DIFFERENTIAL EQUATIONS

4 DEGENERATE or INFLECTED NODE Equal eigenvalues, but one corresponding eigenvector.

5 SPIRAL or FOCUS Two complex eigenvalues with either negative real part STABLE or positive real part UNSTABLE.

6 CENTER Two pure imaginary eigenvalues.

1.6. CRITICAL POINTS AND STABILITY

1.6

51

Critical Points and Stability

1.6.1

Critical Points

In all the systems we have looked at so far (linear, constant coefficient, homogeneous 2-d systems) the origin has been one trajectory all on its own, because if you start at the origin y = 0 you stay there. If

y˙ = Ay

then

y = 0 =⇒ y˙ = 0.

It is the so called trivial solution we mentioned before. In fact any point in Phase Space where y˙ = 0 must be stationary and it is often called a stationary, e...