DIFFERENTIAL EQUATIONS Paul Dawkins, Notas de estudo de Engenharia Civil

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DIFFERENTIAL

EQUATIONS

Paul Dawkins

© 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx

© 2007 Paul Dawkins iii http://tutorial.math.lamar.edu/terms.aspx

Preface

Here are my online notes for my differential equations course that I teach here at Lamar University. Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn how to solve differential equations or needing a refresher on differential equations.

I’ve tried to make these notes as self contained as possible and so all the information needed to read through them is either from a Calculus or Algebra class or contained in other sections of the notes.

A couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn differential equations I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. You will need to find one of your fellow class mates to see if there is something in these notes that wasn’t covered in class.
2. In general I try to work problems in class that are different from my notes. However, with Differential Equation many of the problems are difficult to make up on the spur of the moment and so in this class my class work will follow these notes fairly close as far as worked problems go. With that being said I will, on occasion, work problems off the top of my head when I can to provide more examples than just those in my notes. Also, I often don’t have time in class to work all of the problems in the notes and so you will find that some sections contain problems that weren’t worked in class due to time restrictions.
3. Sometimes questions in class will lead down paths that are not covered here. I try to anticipate as many of the questions as possible in writing these up, but the reality is that I can’t anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that I’ve not included here. You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are.
4. This is somewhat related to the previous three items, but is important enough to merit its own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Using these notes as a substitute for class is liable to get you in trouble. As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class.

© 2007 Paul Dawkins iv http://tutorial.math.lamar.edu/terms.aspx

Outline

Here is a listing and brief description of the material in this set of notes.

Basic Concepts Definitions – Some of the common definitions and concepts in a differential equations course Direction Fields – An introduction to direction fields and what they can tell us about the solution to a differential equation. Final Thoughts – A couple of final thoughts on what we will be looking at throughout this course.

First Order Differential Equations Linear Equations – Identifying and solving linear first order differential equations. Separable Equations – Identifying and solving separable first order differential equations. We’ll also start looking at finding the interval of validity from the solution to a differential equation. Exact Equations – Identifying and solving exact differential equations. We’ll do a few more interval of validity problems here as well. Bernoulli Differential Equations – In this section we’ll see how to solve the Bernoulli Differential Equation. This section will also introduce the idea of using a substitution to help us solve differential equations. Substitutions – We’ll pick up where the last section left off and take a look at a couple of other substitutions that can be used to solve some differential equations that we couldn’t otherwise solve. Intervals of Validity – Here we will give an in-depth look at intervals of validity as well as an answer to the existence and uniqueness question for first order differential equations. Modeling with First Order Differential Equations – Using first order differential equations to model physical situations. The section will show some very real applications of first order differential equations. Equilibrium Solutions – We will look at the behavior of equilibrium solutions and autonomous differential equations. Euler’s Method – In this section we’ll take a brief look at a method for approximating solutions to differential equations.

Second Order Differential Equations Basic Concepts – Some of the basic concepts and ideas that are involved in solving second order differential equations. Real Roots – Solving differential equations whose characteristic equation has real roots. Complex Roots – Solving differential equations whose characteristic equation complex real roots.

© 2007 Paul Dawkins vi http://tutorial.math.lamar.edu/terms.aspx

determinant of a matrix, linearly independent vectors and systems of equations revisited. Review : Eigenvalues and Eigenvectors – Finding the eigenvalues and eigenvectors of a matrix. This topic will be key to solving systems of differential equations. Systems of Differential Equations – Here we will look at some of the basics of systems of differential equations. Solutions to Systems – We will take a look at what is involved in solving a system of differential equations. Phase Plane – A brief introduction to the phase plane and phase portraits. Real Eigenvalues – Solving systems of differential equations with real eigenvalues. Complex Eigenvalues – Solving systems of differential equations with complex eigenvalues. Repeated Eigenvalues – Solving systems of differential equations with repeated eigenvalues. Nonhomogeneous Systems – Solving nonhomogeneous systems of differential equations using undetermined coefficients and variation of parameters. Laplace Transforms – A very brief look at how Laplace transforms can be used to solve a system of differential equations. Modeling – In this section we’ll take a quick look at some extensions of some of the modeling we did in previous sections that lead to systems of equations.

Series Solutions Review : Power Series – A brief review of some of the basics of power series. Review : Taylor Series – A reminder on how to construct the Taylor series for a function. Series Solutions – In this section we will construct a series solution for a differential equation about an ordinary point. Euler Equations – We will look at solutions to Euler’s differential equation in this section.

Higher Order Differential Equations Basic Concepts for n th^ Order Linear Equations – We’ll start the chapter off with a quick look at some of the basic ideas behind solving higher order linear differential equations. Linear Homogeneous Differential Equations – In this section we’ll take a look at extending the ideas behind solving 2nd^ order differential equations to higher order. Undetermined Coefficients – Here we’ll look at undetermined coefficients for higher order differential equations. Variation of Parameters – We’ll look at variation of parameters for higher order differential equations in this section. Laplace Transforms – In this section we’re just going to work an example of using Laplace transforms to solve a differential equation on a 3rd^ order differential equation just so say that we looked at one with order higher than 2nd. Systems of Differential Equations – Here we’ll take a quick look at extending the ideas we discussed when solving 2 x 2 systems of differential equations to systems of size 3 x 3.

© 2007 Paul Dawkins vii http://tutorial.math.lamar.edu/terms.aspx

Series Solutions – This section serves the same purpose as the Laplace Transform section. It is just here so we can say we’ve worked an example using series solutions for a differential equations of order higher than 2nd.

Boundary Value Problems & Fourier Series Boundary Value Problems – In this section we’ll define the boundary value problems as well as work some basic examples. Eigenvalues and Eigenfunctions – Here we’ll take a look at the eigenvalues and eigenfunctions for boundary value problems. Periodic Functions and Orthogonal Functions – We’ll take a look at periodic functions and orthogonal functions in section. Fourier Sine Series – In this section we’ll start looking at Fourier Series by looking at a special case : Fourier Sine Series. Fourier Cosine Series – We’ll continue looking at Fourier Series by taking a look at another special case : Fourier Cosine Series. Fourier Series – Here we will look at the full Fourier series. Convergence of Fourier Series – Here we’ll take a look at some ideas involved in the just what functions the Fourier series converge to as well as differentiation and integration of a Fourier series.

Partial Differential Equations The Heat Equation – We do a partial derivation of the heat equation in this section as well as a discussion of possible boundary values. The Wave Equation – Here we do a partial derivation of the wave equation. Terminology – In this section we take a quick look at some of the terminology used in the method of separation of variables. Separation of Variables – We take a look at the first step in the method of separation of variables in this section. This first step is really the step motivates the whole process. Solving the Heat Equation – In this section we go through the complete separation of variables process and along the way solve the heat equation with three different sets of boundary conditions. Heat Equation with Non-Zero Temperature Boundaries – Here we take a quick look at solving the heat equation in which the boundary conditions are fixed, non-zero temperature conditions. Laplace’s Equation – We discuss solving Laplace’s equation on both a rectangle and a disk in this section. Vibrating String – Here we solve the wave equation for a vibrating string. Summary of Separation of Variables – In this final section we give a quick summary of the method of separation of variables.

© 2007 Paul Dawkins 2 http://tutorial.math.lamar.edu/terms.aspx

Definitions

Differential Equation The first definition that we should cover should be that of differential equation. A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives.

There is one differential equation that everybody probably knows, that is Newton’s Second Law of Motion. If an object of mass m is moving with acceleration a and being acted on with force F then Newton’s Second Law tells us.

F = ma (1)

To see that this is in fact a differential equation we need to rewrite it a little. First, remember that we can rewrite the acceleration, a , in one of two ways. 2

OR 2

dv d u

a a

dt dt

Where v is the velocity of the object and u is the position function of the object at any time t. We should also remember at this point that the force, F may also be a function of time, velocity, and/or position.

So, with all these things in mind Newton’s Second Law can now be written as a differential equation in terms of either the velocity, v , or the position, u , of the object as follows.

( , )

dv

m F t v

dt

2

2 ,^ ,

d u du

m F t u

dt dt

Ê ˆ

= Á ˜

Ë ¯

So, here is our first differential equation. We will see both forms of this in later chapters.

Here are a few more examples of differential equations.

ay ¢¢ + by ¢+ cy = g t ( ) (5)

( ) ( )

2 2 5

sin 2 1

y d y^ y dy y y

dx dx

= - + e^ - (6)

y^ (^4 )^ + 10 y ¢¢¢ - 4 y ¢+ 2 y = cos( t ) (7) 2 2 2

u u

x t

a

2

a u xx = utt (9)

3

u u

x t y

Order The order of a differential equation is the largest derivative present in the differential equation. In the differential equations listed above (3) is a first order differential equation, (4), (5), (6), (8),

© 2007 Paul Dawkins 3 http://tutorial.math.lamar.edu/terms.aspx

and (9) are second order differential equations, (10) is a third order differential equation and (7) is a fourth order differential equation.

Note that the order does not depend on whether or not you’ve got ordinary or partial derivatives in the differential equation.

We will be looking almost exclusively at first and second order differential equations in these notes. As you will see most of the solution techniques for second order differential equations can be easily (and naturally) extended to higher order differential equations and we’ll discuss that idea later on.

Ordinary and Partial Differential Equations A differential equation is called an ordinary differential equation , abbreviated by ode, if it has ordinary derivatives in it. Likewise, a differential equation is called a partial differential equation , abbreviated by pde, if it has differential derivatives in it. In the differential equations above (3) - (7) are ode’s and (8) - (10) are pde’s.

The vast majority of these notes will deal with ode’s. The only exception to this will be the last chapter in which we’ll take a brief look at a common and basic solution technique for solving pde’s.

Linear Differential Equations A linear differential equation is any differential equation that can be written in the following form.

an ( t ) y (^^ n )^^ ( t ) + an (^) - 1 ( t ) y (^ n -^1 )^ ( t ) + L + a 1 (^) ( t ) y ¢( t ) + a 0 ( t ) y t ( ) = g t ( ) (11)

The important thing to note about linear differential equations is that there are no products of the

function, y t ( (^) ), and its derivatives and neither the function or its derivatives occur to any power

other than the first power.

The coefficients a 0 (^) ( t (^) ) , K , an (^) ( t )and g t ( (^) )can be zero or non-zero functions, constant or non-

constant functions, linear or non-linear functions. Only the function, y t ( (^) ), and its derivatives are

used in determining if a differential equation is linear.

If a differential equation cannot be written in the form, (11) then it is called a non-linear differential equation.

In (5) - (7) above only (6) is non-linear, the other two are linear differential equations. We can’t classify (3) and (4) since we do not know what form the function F has. These could be either linear or non-linear depending on F.

Solution

A solution to a differential equation on an interval a < t < b is any function y t ( (^) )which

satisfies the differential equation in question on the interval a < t < b. It is important to note

that solutions are often accompanied by intervals and these intervals can impart some important information about the solution. Consider the following example.

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I’ll leave the details to you to check that these are in fact solutions. Given these examples can you come up with any other solutions to the differential equation? There are in fact an infinite number of solutions to this differential equation.

So, given that there are an infinite number of solutions to the differential equation in the last example (provided you believe me when I say that anyway….) we can ask a natural question. Which is the solution that we want or does it matter which solution we use? This question leads us to the next definition in this section.

Initial Condition(s) Initial Condition(s) are a condition, or set of conditions, on the solution that will allow us to determine which solution that we are after. Initial conditions (often abbreviated i.c.’s when I’m feeling lazy…) are of the form,

( 0 ) 0 and/or^ (^ )^ ( 0 )

k

y t = y y t = y k

So, in other words, initial conditions are values of the solution and/or its derivative(s) at specific points. As we will see eventually, solutions to “nice enough” differential equations are unique and hence only one solution will meet the given conditions.

The number of initial conditions that are required for a given differential equation will depend upon the order of the differential equation as we will see.

Example 2 (^) ( )

3 y x = x -^2 is a solution to 4 x y^2 ¢¢ + 12 xy ¢+ 3 y = 0 , (^) ( 4 ) 1

y = , and

( )

y ¢^ = -.

Solution As we saw in previous example the function is a solution and we can then note that

( ) ( )

( ) ( )

3 2 3 5 2 5

y

y

and so this solution also meets the initial conditions of y ( 4 ) = 18 and y ¢^ ( 4 ) = - 643. In fact,

( )

3

y x = x -^2 is the only solution to this differential equation that satisfies these two initial

conditions.

Initial Value Problem An Initial Value Problem (or IVP ) is a differential equation along with an appropriate number of initial conditions.

Example 3 The following is an IVP.

( ) ( )

x y ¢¢^ + xy ¢^ + y = y = y ¢ = -

© 2007 Paul Dawkins 6 http://tutorial.math.lamar.edu/terms.aspx

Example 4 Here’s another IVP.

2 t y ¢^ + 4 y = 3 y ( 1 ) = - 4

As I noted earlier the number of initial condition required will depend on the order of the differential equation.

Interval of Validity The interval of validity for an IVP with initial condition(s)

( 0 ) 0 and/or^ (^ )^ ( 0 )

k

y t = y y t = y k

is the largest possible interval on which the solution is valid and contains t 0. These are easy to

define, but can be difficult to find, so I’m going to put off saying anything more about these until we get into actually solving differential equations and need the interval of validity.

General Solution The general solution to a differential equation is the most general form that the solution can take and doesn’t take any initial conditions into account.

Example 5 (^) ( ) (^2)

c

y t

t

= + is the general solution to

2 t y ¢^ + 4 y = 3

I’ll leave it to you to check that this function is in fact a solution to the given differential equation. In fact, all solutions to this differential equation will be in this form. This is one of the first differential equations that you will learn how to solve and you will be able to verify this shortly for yourself.

Actual Solution The actual solution to a differential equation is the specific solution that not only satisfies the differential equation, but also satisfies the given initial condition(s).

Example 6 What is the actual solution to the following IVP?

2 t y ¢ + 4 y = 3 y ( 1 ) = - 4

Solution This is actually easier to do than it might at first appear. From the previous example we already know (well that is provided you believe my solution to this example…) that all solutions to the differential equation are of the form.

( ) (^2)

c

y t

t

All that we need to do is determine the value of c that will give us the solution that we’re after. To find this all we need do is use our initial condition as follows.

( ) (^2)

c

• = y = + fi c = - - = -

So, the actual solution to the IVP is.

( ) (^2)

y t

t

© 2007 Paul Dawkins 8 http://tutorial.math.lamar.edu/terms.aspx

Direction Fields

This topic is given its own section for a couple of reasons. First, understanding direction fields and what they tell us about a differential equation and its solution is important and can be introduced without any knowledge of how to solve a differential equation and so can be done here before we get into solving them. So, having some information about the solution to a differential equation without actually having the solution is a nice idea that needs some investigation.

Next, since we need a differential equation to work with this is a good section to show you that differential equations occur naturally in many cases and how we get them. Almost every physical situation that occurs in nature can be described with an appropriate differential equation. The differential equation may be easy or difficult to arrive at depending on the situation and the assumptions that are made about the situation and we may not ever be able to solve it, however it will exist.

The process of describing a physical situation with a differential equation is called modeling. We will be looking at modeling several times throughout this class.

One of the simplest physical situations to think of is a falling object. So let’s consider a falling object with mass m and derive a differential equation that, when solved, will give us the velocity of the object at any time, t. We will assume that only gravity and air resistance will act upon the object as it falls. Below is a figure showing the forces that will act upon the object.

Before defining all the terms in this problem we need to set some conventions. We will assume that forces acting in the downward direction are positive forces while forces that act in the upward direction are negative. Likewise, we will assume that an object moving downward ( i.e. a falling object) will have a positive velocity.

Now, let’s take a look at the forces shown in the diagram above. FG is the force due to gravity

and is given by FG = mg where g is the acceleration due to gravity. In this class I use g = 9.

m/s^2 or g = 32 ft/s^2 depending on whether we will use the metric or British system. FA is the

force due to air resistance and for this example we will assume that it is proportional to the

velocity, v , of the mass. Therefore the force due to air resistance is then given by FA = - g v ,

where g > 0. Note that the “–” is required to get the correct sign on the force. Both g and v are

positive and the force is acting upward and hence must be negative. The “–” will give us the correct sign and hence direction for this force.

Recall from the previous section that Newton’s Second Law of motion can be written as

( , )

dv

m F t v

dt

where F(t,v) is the sum of forces that act on the object and may be a function of the time t and the velocity of the object, v. For our situation we will have two forces acting on the object gravity,

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F G = mg. acting in the downward direction and hence will be positive, and air resistance,

F A = - g v , acting in the upward direction and hence will be negative. Putting all of this together

into Newton’s Second Law gives the following.

dv

m mg v

dt

= - g

To simplify the differential equation let’s divide out the mass, m.

dv v

g

dt m

g

This then is a first order linear differential equation that, when solved, will give the velocity, v (in m/s), of a falling object of mass m that has both gravity and air resistance acting upon it.

In order to look at direction fields (that is after all the topic of this section....) it would be helpful to have some numbers for the various quantities in the differential equation. So, let’s assume that we have a mass of 2 kg and that g = 0.392. Plugging this into (1) gives the following differential equation.

dv

v

dt

Let's take a geometric view of this differential equation. Let's suppose that for some time, t , the velocity just happens to be v = 30 m/s. Note that we’re not saying that the velocity ever will be 30 m/s. All that we’re saying is that let’s suppose that by some chance the velocity does happen to be 30 m/s at some time t. So, if the velocity does happen to be 30 m/s at some time t we can plug v = 30 into (2) to get.

dv

dt

Recall from your Calculus I course that a positive derivative means that the function in question, the velocity in this case, is increasing, so if the velocity of this object is ever 30m/s for any time t the velocity must be increasing at that time.

Also, recall that the value of the derivative at a particular value of t gives the slope of the tangent line to the graph of the function at that time, t. So, if for some time t the velocity happens to be 30 m/s the slope of the tangent line to the graph of the velocity is 3.92.

We could continue in this fashion and pick different values of v and compute the slope of the tangent line for those values of the velocity. However, let's take a slightly more organized approach to this. Let's first identify the values of the velocity that will have zero slope or horizontal tangent lines. These are easy enough to find. All we need to do is set the derivative equal to zero and solve for v.

In the case of our example we will have only one value of the velocity which will have horizontal tangent lines, v = 50 m/s. What this means is that IF (again, there’s that word if), for some time t , the velocity happens to be 50 m/s then the tangent line at that point will be horizontal. What the slope of the tangent line is at times before and after this point is not known yet and has no bearing on the slope at this particular time, t.

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Now, let’s look at v > 50. The first thing to do is to find out if the slopes are positive or negative. We will do this the same way that we did in the last bit, i.e. pick a value of v , plug this into (2) and see if the derivative is positive or negative. Note, that you should NEVER assume that the derivative will change signs where the derivative is zero. It is easy enough to check so you should always do so.

We need to check the derivative so let's use v = 60. Plugging this into (2) gives the slope of the tangent line as -1.96, or negative. Therefore, for all values of v > 50 we will have negative slopes for the tangent lines. As with v < 50, by looking at (2) we can see that as v approaches 50, always staying greater than 50, the slopes of the tangent lines will approach zero and flatten out. While moving v away from 50 again, staying greater than 50, the slopes of the tangent lines will become steeper. We can now add in some arrows for the region above v = 50 as shown in the graph below.

This graph above is called the direction field for the differential equation.

So, just why do we care about direction fields? There are two nice pieces of information that can be readily found from the direction field for a differential equation.

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1. Sketch of solutions. Since the arrows in the direction fields are in fact tangents to the actual solutions to the differential equations we can use these as guides to sketch the graphs of solutions to the differential equation.
2. Long Term Behavior. In many cases we are less interested in the actual solutions to the differential equations as we are in how the solutions behave as t increases. Direction fields, if we can get our hands on them, can be used to find information about this long term behavior of the solution.

So, back to the direction field for our differential equation. Suppose that we want to know what the solution that has the value v (0) = 30 looks like. We can go to our direction field and start at 30 on the vertical axis. At this point we know that the solution is increasing and that as it increases the solution should flatten out because the velocity will be approaching the value of v = 50. So we start drawing an increasing solution and when we hit an arrow we just make sure that we stay parallel to that arrow. This gives us the figure below.

To get a better idea of how all the solutions are behaving, let's put a few more solutions in. Adding some more solutions gives the figure below. The set of solutions that we've graphed below is often called the family of solution curves or the set of integral curves. The number of solutions that is plotted when plotting the integral curves varies. You should graph enough solution curves to illustrate how solutions in all portions of the direction field are behaving.