Differential Equations: Existence, Uniqueness, and Portraits for First-Order Equations, Schemes and Mind Maps of Differential Equations

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Differential

Equations

by

R. Decker and B. Albright

• 1 Introduction to Differential Equations
  • 1.1 A Brief Overview of Differential Equations
  • 1.2 Explicit, Numerical, and Graphical Solutions
  • 1.3 Mathematical Modeling with Differential Equations: General Principles
  • 1.4 Deriving Models of Electrical Circuits
  • 1.5 Using Lagrange’s Equations to Model Mechanical Systems
• 2 First-order Differential Equations
  • 2.1 Linear first-order differential equations
  • 2.2 Existence, uniqueness, and portraits for first-order equations
  • 2.3 Separable Differential Equations
  • 2.4 Numerical methods for first-order equations
  • 2.5 Autonomous first-order equations and bifurcations
• 3 Second-order Differential Equations
  • 3.1 Linear Second-order Equations
  • 3.2 Harmonic Oscillators
  • 3.3 Nonhomogeneous Equations and The Method of Undetermined Coefficients
  • 3.4 Driven Mass-Spring Systems, Beats, and Resonance
  • 3.5 Numerical Methods for Second-Order DE’s and Systems
  • 3.6 Qualitative Methods and the Phase Plane ii CONTENTS
• 4 Linear Algebra Interlude
  • 4.1 Matrices, Vectors, Scalars
  • 4.2 Eigenvalues and Eigenvectors
• 5 Systems of First-order Differential Equations
  • 5.1 Explicit Solutions of Constant-coefficient Linear Systems
  • 5.2 Stability of Autonomous Linear Systems
  • 5.3 Fixed Points and Stability of Nonlinear Autonomous Systems
  • 5.4 Bifurcations in Systems
• 6 Laplace Transforms
  • 6.1 Simple Laplace Transforms
  • 6.2 Deriving More Laplace Transforms
  • 6.3 The Unit Step and Delta functions
  • 6.4 Convolution and Circuits

Preface

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The classical approach to introductory differential equations textbooks is to present tech- niques for analytically solving different categories of differential equations and then ana- lyzing the solutions algebraically. In this book we take a more modern approach, utilizing software to graphically and numerically solve differential equations. The focus of this text is on the setting up, or modeling, of the equations and the analysis of their solutions.

This text is intended for a one semester introduction course to differential equations for math, science, and engineering majors. The prerequisite is two semesters of calculus. This book is intended to be used with available software such as Maple, Mathematica, Mat- lab, Maxima, Wolfram Alpha, TI CAS enabled calculators, and websites. Interactive java graphing applets for first-order differential equations andfirst-order systems of two equations are available at uhaweb.hartford.edu/rdecker/DeckerDEbook/DeckerDEbook.html (no www at the beginning). Other applets specifically related to examples in the text are located there also.

Chapter 1

Introduction to Differential

Equations

In this chapter we introduce the main concepts behind differential equations, why they are important, how they can be derived (created), and how information can be extracted from them in the form of various types of solutions (exact, graphical and numerical). The rest of the text will develop these ideas further by categorizing differential equations and introducing techniques specific to those categories.

1.1 A Brief Overview of Differential Equations

The physical laws of the universe are written in the language of differential equations. The classical mechanics of Newton, Lagrange and Hamilton, the fluid mechanics of Bernoulli and Euler, and Maxwell’s theory of electricity and magnetism are all expressed via differential equations - and form much of the theoretical basis of the engineering disciplines. In the area of modern physics, Einstein’s theory of general relativity and the quantum mechanics of Schrodinger and Dirac are based on differential equations. Differential equations have invaded many other branches of science, including (but not limited to) chemistry, biology, economics and finance, and meteorology. It is no exaggeration to claim that the modern world as we know it could not have come into being without the development of this branch of mathematics.

3

4 CHAPTER 1 Introduction to Differential Equations

Laplace’s dream

The beginning student may be surprised to find that differential equations can be used to predict the future - and they have a much better track record than any psychic. To quote the great mathematician Pierre-Simon Laplace^1

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. —Pierre Simon Laplace, A Philosophical Essay on Probabilities^2

The engineer and the crumpled paper

The process described by Laplace goes something like this. Imagine that an engineer has accidently knocked a crumpled piece of paper out of her third story window, 8 meters above the ground. Being a good environmentalist, she wonders how much time she has before the paper hits the ground. She immediately recalls Newton’s second law F = ma. The law says that the sum of all of the forces acting on a body are equal to the mass of the body multiplied by its acceleration. There are two forces acting on the paper; the force of gravity (pulling it downward) and the force of air resistance (which acts in the opposite direction of the motion).

Working furiously, she assigns the variable x to the position of the paper (distance above the ground in meters), lets t (in seconds) represent the time elapsed since she dropped the paper, and recalls from calculus that acceleration is the second derivative of position. Newton’s second law becomes F = mx′′. She also knows that the force of gravity is given by mg where m is the mass of the body and g is the acceleration due to gravity (in meters per second^2 ).

(^1) Laplace, Pierre Simon, A Philosophical Essay on Probabilities, translated into English from the original French 6th ed. by Truscott, F.W. and Emory, F.L., Dover Publications (New York, 1951) p. (^2) Due to the development of quantum theory in the 1920’s and 1930’s,this statement must be modified a bit - the differential equations of quantum mechanics make predictions about the probabilities of certain events occurring, at least on a microscopic scale. The spirit of the statement still holds, as extremely accurate predictions of such probabilities can be made.

6 CHAPTER 1 Introduction to Differential Equations

Using her knowledge of differential equations, she obtains the following solution:

x(t) = 2 :45 exp( 2 t) 4 : 9 t + 10: 45

This function predicts the height above the ground of the crumpled paper for any value of t (in seconds). One can easily verify that this function solves the differential equation and satisfies the initial conditions (this will be done later). This function predicts that after 1 second, for instance, the height of the paper is

x(1) = 2 :45 exp(2(1)) 4 :9(1) + 10: 45  5 : 22 meters.

The original question is, “When does the paper hit the ground?” At the time the paper hits the ground, the height is 0 meters. So to determine the time the paper hits the ground she needs to solve the equation

2 :45 exp( 2 t) 4 : 9 t + 10:45 = 0;

which cannot be solved algebraically. A numerical method of solution, such as Newton’s method from calculus, is required. Such numerical solutions are built into Computer Al- gebra System (CAS) software. Using available software she quickly obtains the solution t = 2: 126 seconds, accurate to three decimal places.

The entire process described above has taken the engineer only about a second (if you doubt this is possible, just watch any episode of the television show “Numb3rs”). Thus she still has time to save the ground from litter. She leans out the window and shouts to a bystander below, “Could you please catch that piece of paper for me?” At precisely 2 : 1 seconds after the paper began its fall, and with just 0 : 026 seconds to spare, the bystander reaches out and grabs the crumpled paper, thus saving the world from one more piece of litter. See Figure 1.1 for a graphical visualization of this solution.

Terminology

We begin with a definition of a differential equation.

De nition 1.1.1 A differential equation (or DE for short) is an equation containing

some or all of the following: an unknown function, the independent variable of

1.1 A Brief Overview of Differential Equations 7

Figure 1.1: The falling crumpled paper

the unknown function, at least one derivative of the function, and one or more constants. The constants are also called parameters , and the function itself is called the dependent variable.

Every differential equation contains at least one derivative of the unknown function. These derivatives could be the first derivative, the second derivative, etc., or any combination of these. This leads to the definition of the order of a differential equation.

De nition 1.1.2 The order of a differential equation is the number of the highest

derivative that appears in the equation.

Example 1.1.1 The following examples help illustrate these definitions:

Differential Dependent Independent Equation Variable Variable Parameter(s) Order

1. x(t)′′^ = (^) mc x(t)′^ g x t c, m, g 2
2. y′′^ + 4y = sin(t) y t none 2
3. y′^ + ay = ex^ y x a 1
4. x′^ = ax(1 x=b) x t a, b 1

1.1 A Brief Overview of Differential Equations 9

Then we substitute these derivatives into the DE and verify that both sides really are equal:

9 :8 exp( 2 t) =? 2 (4:9 exp( 2 t) 4 :9) 9 : 8

9 :8 exp( 2 t)

9 :8 exp( 2 t) + 9: 8 9 : 8 9 :8 exp( 2 t) =^ 9 :8 exp( 2 t):

Note that we put question marks over the equal signs because we are not sure that the two sides really are equal, until the last equation. Since this last equation is indeed true, we conclude that x(t) = 2 :45 exp( 2 t) 4 : 9 t + 10: 45 is a solution.

Verifying that a given function is a solution is relatively easy. Finding a solution is an- other issue. Sometimes, as illustrated in the next two examples, we can find a solution by reasoning with our knowledge of calculus and making educated guesses.

Example 1.1.3 Find a solution to the DE y′^ = y.

Solution. Note that the differential equation expressed in words says “the derivative of a function is equal to function you started with.” We should ask ourselves if we know of function with this property. From calculus we know that the only function that is its own derivative is the exponential function y = ex:

We now check our guess. The derivative of of y = ex^ is y′^ = ex. Substituting ex^ in for both y and y′^ in the differential equation y′^ = y we get

ex^ = ex;

which is clearly a true statement. This verifies our solution.

In the previous example, it may have been a surprise to see an exponential function appear as the solution to a differential equation that itself had no exponential function in it. It is often the case that a solution to differential equations looks nothing like the DE from which it came.

Example 1.1.4 Find a solution to the DE y′′^ = y.

Solution. In words, this DE says, “the second derivative of a function is equal to the negative of the function.” We might consider the exponential function y = ex, but this does not work this time, as its second derivative is ex, not ex. To get a negative sign involved, we might try y = ex^ But then y′^ = ex^ and y′′^ = ex^ which is the original function and not the negative of it.

10 CHAPTER 1 Introduction to Differential Equations

So let’s consider something completely different. Consider the trigonometric function y = cos(x). Its first and second derivatives are

y′^ = sin(x) and y′′^ = cos(x):

Substituting these derivatives into the DE y′′^ = y we get cos(x) = cos(x), which is a true statement. This verifies the solution. Similar calculations verify that y = sin(x) also is a solution.

Comparison of algebraic equations and differential equations

A differential equation is an equation involving an uknown function. In the equations we solved in high school algebra, the unknown was a number. Such equations are called algebraic equations. For example, the equation

2 x + 1 = 7

is an algebraic equation. To solve this equation, we “do the opposite of what is being done to the unknown” by subtracting 1 from both sides, and then dividing by 2 to get x = 3. To check this solution, we substitute 3 in for x in the equation 2 x + 1 = 7 to get 2(3) + 1 = 7, which is a true statement.

A solution to an algebraic equation is a number , whereas the solution to a differential equation is a function. To check a solution both cases we substitute the claimed solution into the equation, and if we get a true statement then we have shown that the solution is correct. With simple algebraic equations we may be able to guess the solution, but for more complicated ones we need algebra. Differential equations are similar. For very simple DE’s, as the ones we encountered in the previous two examples, we were able to guess a solution. But more generally we will need some techniques (using calculus in addition to algebra) for coming up with solutions. Throughout this text we will develop techniques for doing just this.

Solving a DE problem can involve solving both DE’s and algebraic equations. Consider the crumpled paper scenario. To find when the paper hit the ground, we first had to solve the DE x′′^ = 2 x′^ 9 : 8 ;

and then we had to solve the algebraic equation

2 :45 exp( 2 t) 4 : 9 t + 10:45 = 0:

12 CHAPTER 1 Introduction to Differential Equations

De nition 1.1.4 A general solution of a first order DE is a solution with one

constant that does not appear in the original differential equation. This is also called a one-parameter family of solutions. When a particular value of the constant is chosen, we get a particular solution.

In some texts, the term general solution is reserved only for those situations where every possible solution to the differential equation corresponds to some value of the arbitrary constant. In this text, we use the term general solution in a slightly different way. General solutions to higher-order equations and systems of equations will be discussed in later chapters.

Not all general solutions involve a constant added to the end, as illustrated in the next example.

Example 1.1.5 Show that y = 2ex, y = 3 ex, and y = 0 are all solutions to the DE y′^ = y

and find a general solution to this DE.

Solution. The derivatives of these claimed solutions are 2 ex, 3 ex, and 0 , respectively. Notice that in each case, the derivative equals the function. Thus these are indeed solutions.

To find a general solution, note that all these solutions are of the form Cex^ where C is a constant (in the solution y = 0, the constant is C = 0). Therefore, a general solution is y = Cex.

We can picture a general solution to a first-order differential equation by graphing the solution for several values of the constant. Such graphs are called solution curves. Figure 1.2 shows the solution curves to y′^ = y corresponding to C = 2 , 1 , 0 , 1 , and 2.

The next example illustrates that a general solution does not always describe all of the solutions to a DE.

Example 1.1.6 Show that a general solution to the differential equation P ′^ = P 2 is

P =

C + t

Also show that P = 0 is also a solution that does not correspond to any real value of C.

1.1 A Brief Overview of Differential Equations 13

Figure 1.2: Graph of y = Cex^ for C = 2 ; 1 ; 0 ; 1 ; 2

Solution. Using the quotient rule, the derivative of the claimed solution is

P ′^ =

(C + t)
0 (1)
1 (C + t)^2

(C + t)^2

Substituting P and its derivative into the DE P ′^ = P 2 yields

1 (C + t)^2

C + t

which is a true statement. This verifies the solution. Note that P = 0 is also a solution because the derivative of this solution is P ′^ = 0. Thus P ′^ = P is a true statement. However, no value of C will make 1 C + t

= 0 for all t:

(One might argue that C = 1 works, but 1 is not a real number.) Thus the solution P = 0 is not “part of” the general solution P = 1 =(C + t).

1.1 A Brief Overview of Differential Equations 15

of populations (animal, human, plant, microbial, etc.). Though such models may lack the predictive precision of Newton’s second law, they can be very useful to scientists trying to manage natural resources.

The simplest population model is based on the assumption that the rate of growth of the population is proportional to the size of the population. This assumption is simply saying that large populations grow at a faster rate than small populations. If y(t) represents the number of organisms at time t, then this assumption yields the differential equation

y′^ = ky

where k is a constant called the growth constant or growth rate.

Suppose a population of bacteria in a Petri dish grows according to this DE with growth constant k = 3, where time t is in days and y(t) is measured in hundreds of bacteria. Suppose that there were initially 1,000 organisms in the dish. Show that y = Ce^3 t^ is a general solution of this DE, and use the initial condition to predict how many bacteria there will be in 2 days.

Solution. To verify the general solution, note that its derivative is

y′^ = 3Ce^3 t:

Substituting the claimed solution and its derivative into the DE y′^ = 3y, we get 3 Ce^3 t^ on both sides, verifying that y = Ce^3 t^ is a general solution.

If we take the initial time to be t = 0, then the initial condition is y(0) = 10. To find C, we substitute t = 0 and y = 10 into y = Ce^3 t^ to get

10 = Ce3(0)^ = C;

which yields the particular solution y = 10e^3 t. This solution can be used to make predic- tions. After t = 2 days we predict that we will have

y = 10e3(2)^  4034 : 3 thousand bacteria:

We must stress that such a prediction is at best an approximation, based on the assumption that the population is described by the DE y′^ = 3y. If this assumption is not accurate, then our prediction is not accurate, regardless of the rigor of our mathematics.

16 CHAPTER 1 Introduction to Differential Equations

Higher order differential equations

Our discussion of general solutions, arbitrary constants, and initial conditions has so far been restricted to first-order differential equations. To summarize, for a first-order differ- ential equation, the general solution will contain one arbitrary constant, and we therefore need one initial condition in order to determine that constant.

In our discussion of general solutions of first-order DE’s, we saw that the arbitrary con- stant comes from integrating to “undo” the derivative. A second-order DE involves two derivatives. So, informally, we will have to integrate twice to solve it. This results in two arbitrary constants in a general solution. Finding the values of these constants requires the use of two initial conditions.

Generically, a general solution to an nth^ order DE will contain n arbitrary constants and will require n initial conditions to find a particular solution. These initial conditions can be conditions on the value of the unknown function or on the value(s) of its derivative(s). The next example illustrates this idea.

Example 1.1.9 In the crumpled paper scenario, we found a particular solution to the

second-order DE x′′^ = 2 x′^ 9 : 8. Show that x(t) = 0 : 5 C 1 e^2 t^ 4 : 9 t + C 2 is a general solution to this DE. Then use the initial conditions x(0) = 8 and x′(0) = 0 to determine C 1 and C 2. Compare this result to the particular solution found by the engineer.

Solution. We take two derivatives of x(t) = 0 : 5 C 1 e^2 t^ 4 : 9 t + C 2 to get

x′(t) = C 1 e^2 t^ 4 and x′′(t) = 2 C 1 e^2 t:

Substituting these into the DE x′′^ = 2 x′^ 9 : 8

2 C 1 e^2 t^ = 2(C 1 e^2 t^ 4 :9) 9 : 8 ;

which is a true statement. This verifies the general solution.

Next we use the initial condition x′(0) = 0 to find C 1 by substituting t = 0 and x′^ = 0 into x′(t) = C 1 e^2 t^ 4 : 9 : 0 = C 1 4 : 9 ) C 1 = 4: 9 :

Finally, we use the initial condition x(0) = 8 to find C 2 by substituting t = 0, x = 8, and C 1 = 4: 9 into x(t) = 0 : 5 C 1 e^2 t^ 4 : 9 t + C 2 :

8 = 0 :5(4:9) + C 2 ) C 2 = 10: 45 :