Differential Equations For Dummies - Softcover

Holzner, Steven

Softcover

ISBN 10: 0470178140 ISBN 13: 9780470178140

Publisher: For Dummies, 2008

Synopsis

The fun and easy way to understand and solve complex equations

Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.

About this title

About the Author

Steven Holzner is an award-winning author of science, math, and technical books. He got his training in differential equations at MIT and at Cornell University, where he got his PhD. He has been on the faculty at both MIT and Cornell University, and has written such bestsellers as Physics For Dummies and Physics Workbook For Dummies.

From the Back Cover

Power your way through ordinary and singular points

Understand differential equations through practical tips and examples

Do differential equations cause you distress? No worries! This friendly guide explains this intimidating subject in plain English, walking you step by step through all the key concepts ― from linear and separable first order differential equations to higher order equations, power series, and Laplace transforms. You'll find plenty of examples to increase your problem-solving skills and a variety of helpful definitions and explanations to conquer even the toughest differential equations.

Discover how to:

• Classify differential equations

• Solve with integrating factors

• Work with coefficients

• Use handy theorems

• Have fun with advanced techniques

• Apply differential equations in real life

Understand differential equations through practical tips and examples

Do differential equations cause you distress? No worries! This friendly guide explains this intimidating subject in plain English, walking you step by step through all the key concepts — from linear and separable first order differential equations to higher order equations, power series, and Laplace transforms. You'll find plenty of examples to increase your problem-solving skills and a variety of helpful definitions and explanations to conquer even the toughest differential equations.

Discover how to:

• Classify differential equations

• Solve with integrating factors

• Work with coefficients

• Use handy theorems

• Have fun with advanced techniques

• Apply differential equations in real life

Excerpt. © Reprinted by permission. All rights reserved.

Differential Equations For Dummies

By Steven Holzner

John Wiley & Sons

Copyright © 2008 Steven Holzner
All right reserved.
ISBN: 978-0-470-17814-0

Chapter One

Welcome to the World of Differential Equations

In This Chapter

* Breaking into the basics of differential equations

* Getting the scoop on derivatives

* Checking out direction fields

* Putting differential equations into different categories

* Distinguishing among different orders of differential equations

* Surveying some advanced methods

It's a tense moment in the physics lab. The international team of high-powered physicists has attached a weight to a spring, and the weight is bouncing up and down.

"What's happening?" the physicists cry. "We have to understand this in terms of math! We need a formula to describe the motion of the weight!"

You, the renowned Differential Equations Expert, enter the conversation calmly. "No problem," you say. "I can derive a formula for you that will describe the motion you're seeing. But it's going to cost you."

The physicists look worried. "How much?" they ask, checking their grants and funding sources. You tell them.

"Okay, anything," they cry. "Just give us a formula."

You take out your clipboard and start writing.

"What's that?" one of the physicists asks, pointing at your calculations.

"That," you say, "is a differential equation. Now all I have to do is to solve it, and you'll have your formula." The physicists watch intently as you do your math at lightning speed.

"I've got it," you announce. "Your formula is y = 10 sin (5t), where y is the weight's vertical position, and t is time, measured in seconds."

"Wow," the physicists cry, "all that just from solving a differential equation?"

"Yep," you say, "now pay up."

Well, you're probably not a renowned differential equations expert - not yet, at least! But with the help of this book, you very well may become one. In this chapter, I give you the basics to get started with differential equations, such as derivatives, direction fields, and equation classifications.

The Essence of Differential Equations

REMEMBER

In essence, differential equations involve derivatives, which specify how a quantity changes; by solving the differential equation, you get a formula for the quantity itself that doesn't involve derivatives.

Because derivatives are essential to differential equations, I take the time in the next section to get you up to speed on them. (If you're already an expert on derivatives, feel free to skip the next section.) In this section, however, I take a look at a qualitative example, just to get things started in an easily digestible way.

Say that you're a long-time shopper at your local grocery store, and you've noticed prices have been increasing with time. Here's the table you've been writing down, tracking the price of a jar of peanut butter:

Month Price

1 $2.40 2 $2.50 3 $2.60 4 $2.70 5 $2.80 6 $2.90

Looks like prices have been going up steadily, as you can see in the graph of the prices in Figure 1-1. With that large of a price hike, what's the price of peanut butter going to be a year from now?

You know that the slope of a line is [DELTA]y/[DELTA]x (that is, the change in y divided by the change in x). Here, you use the symbols [DELTA]p for the change in price and [DELTA]t for the change in time. So the slope of the line in Figure 1-1 is [DELTA]p/[DELTA]t.

Because the price of peanut butter is going up 10 cents every month, you know that the slope of the line in Figure 1-1 is:

[DELTA]p/[DELTA]t = 10/month

The slope of a line is a constant, indicating its rate of change. The derivative of a quantity also gives its rate of change at any one point, so you can think of the derivative as the slope at a particular point. Because the rate of change of a line is constant, you can write:

dp/dt = [DELTA]p/[DELTA]t = 10/month

TIP

In this case, dp/dt is the derivative of the price of peanut butter with respect to time. (When you see the d symbol, you know it's a derivative.)

And so you get this differential equation:

dp/dt = 10/month

The previous equation is a differential equation because it's an equation that involves a derivative, in this case, dp/dt. It's a pretty simple differential equation, and you can solve for price as a function of time like this:

p = 10t + c

In this equation, p is price (measured in cents), t is time (measured in months), and c is an arbitrary constant that you use to match the initial conditions of the problem. (You need a constant, c, because when you take the derivative of 10t + c, you just get 10, so you can't tell whether there's a constant that should be added to 10t - matching the initial conditions will tell you.)

The missing link is the value of c, so just plug in the numbers you have for price and time to solve for it. For example, the cost of peanut butter in month 1 is $2.40, so you can solve for c by plugging in 1 for t and $2.40 for p (240 cents), giving you:

240 = 10 + c

By solving this equation, you calculate that c = 230, so the solution to your differential equation is:

p = 10t + 230

And that's your solution - that's the price of peanut butter by month. You started with a differential equation, which gave the rate of change in the price of peanut butter, and then you solved that differential equation to get the price as a function of time, p = 10t + 230.

Want to see the solution to your differential equation in action? Go for it! Find out what the price of peanut butter is going to be in month 12. Now that you have your equation, it's easy enough to figure out:

p = 10t + 230 10(12) + 230 = 350

As you can see, in month 12, peanut butter is going to cost a steep $3.50, which you were able to figure out because you knew the rate at which the price was increasing. This is how any typical differential equation may work: You have a differential equation for the rate at which some quantity changes (in this case, price), and then you solve the differential equation to get another equation, which in this case related price to time.

TIP

Note that when you substitute the solution (p = 10t + 230) into the differential equation, dp/dt indeed gives you 10 cents per month, as it should.

Derivatives: The Foundation of Differential Equations

REMEMBER

As I mention in the previous section, a derivative simply specifies the rate at which a quantity changes. In math terms, the derivative of a function f(x), which is depicted as df(x)/dx, or more commonly in this book, as f'(x), indicates how f(x) is changing at any value of x. The function f(x) has to be continuous at a particular point for the derivative to exist at that point.

Take a closer look at this concept. The amount f(x) changes in a small distance along the x axis [DELTA]x is:

f(x + [DELTA]x) - f(x)

The rate at which f(x) changes over the change [DELTA]x is:

f(x + [DELTA]x) - f(x)/[DELTA]x

So far so good. Now to get the derivative dy/dx, where y = f(x), you must let [DELTA]x get very small, approaching zero. You can do that with a limiting expression, which you can evaluate as [DELTA]x goes to zero. In this case, the limiting expression is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In other words, the derivative of f(x) is the amount f(x) changes in [DELTA]x, divided by [DELTA]x, as [DELTA]x goes to zero.

I take a look at some common derivatives in the following sections; you'll see these derivatives throughout this book.

Derivatives that are constants

The first type of derivative you'll encounter is when f(x) equals a constant, c. If f(x) = c, then f(x + [DELTA]x) = c also, and f(x + [DELTA]x) - f(x) = 0 (because all these amounts are actually the same), so df(x)/dx = 0. Therefore:

f(x) = c df(x)/dx = 0

How about when f(x) = cx, where c is a constant? In this case, f(x) = cx, and = cx + c [DELTA]x.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Derivatives that are powers

Another type of derivative that pops up is one that includes raising x to the power n. Derivatives with powers work like this:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Raising e to a certain power is always popular when working with differential equations (e is the natural logarithm base, e = 2.7128 ..., and a is a constant):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

And there's also the inverse of ea, which is the natural log, which works like this:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Derivatives involving trigonometry

Now for some trigonometry, starting with the derivative of sin(x):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

And here's the derivative of cos(x):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Derivatives involving multiple functions

The derivative of the sum (or difference) of two functions is equal to the sum (or difference) of the derivatives of the functions (that's easy enough to remember!):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The derivative of the product of two functions is equal to the first function times the derivative of the second, plus the second function times the derivative of the first. For example:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

How about the derivative of the quotient of two functions? That derivative is equal to the function in the denominator times the derivative of the function in the numerator, minus the function in the numerator times the derivative of the function in the denominator, all divided by the square of the function in the denominator:

Seeing the Big Picture with Direction Fields

It's all too easy to get caught in the math details of a differential equation, thereby losing any idea of the bigger picture. One useful tool for getting an overview of differential equations is a direction field, which I discuss in more detail in Chapter 2. Direction fields are great for getting a handle on differential equations of the following form:

dy/dx = f(x,y)

REMEMBER

The previous equation gives the slope of the equation y = f(x) at any point x. A direction field can help you visualize such an equation without actually having to solve for the solution. That field is a two-dimensional graph consisting of many, sometimes hundreds, of short line segments, showing the slope - that is, the value of the derivative - at multiple points. In the following sections, I walk you through the process of plotting and understanding direction fields.

Plotting a direction field

Here's an example to give you an idea of what a direction field looks like. A body falling through air experiences this force:

F = mg - [gamma]v

In this equation, F is the net force on the object, m is the object's mass, g is the acceleration due to gravity (g = 9.8 meters/[sec.sup.2] near the Earth's surface), [gamma] is the drag coefficient (which adds the effect of air friction and is measured in newtons sec/meter), and v is the speed of the object as it plummets through the air.

If you're familiar with physics, consider Newton's second law. It says that F = ma, where F is the net force acting on an object, m is its mass, and a is its acceleration. But the object's acceleration is also dv/dt, the derivative of the object's speed with respect to time (that is, the rate of change of the object's speed). Putting all this together gives you:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now you're back in differential equation territory, with this differential equation for speed as a function of time:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now you can get specific by plugging in some numbers. The acceleration due to gravity, g, is 9.8 meters/[sec.sup.2] near the Earth's surface, and let's say that the drag coefficient is 1.0 newtons sec/meter and the object has a mass of 4.0 kilograms. Here's what you'd get:

dv/dt = 9.8 v/4

REMEMBER

To get a handle on this equation without attempting to solve it, you can plot it as a direction field. To do so you create a two-dimensional plot and add dozens of short line segments that give the slope at those locations (you can do this by hand or with software). The direction field for this equation appears in Figure 1-2. As you can see in the figure, there are dozens of short lines in the graph, each of which give the slope of the solution at that point. The vertical axis is v, and the horizontal axis is t.

TIP

Because the slope of the solution function at any one point doesn't depend on t, the slopes along any horizontal line are the same.

Connecting slopes into an integral curve

REMEMBER

You can get a visual handle on what's happening with the solutions to a differential equation by looking at its direction field. How? All those slanted line segments give you the solutions of the differential equations - all you have to do is draw lines connecting the slopes. One such solution appears in Figure 1-3. A solution like the one in the figure is called an integral curve of the differential equation.

Recognizing the equilibrium value

As you can see from Figure 1-3, there are many solutions to the equation that you're trying to solve. As it happens, the actual solution to that differential equation is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the previous solution, c is an arbitrary constant that can take any value. That means there are an infinite number of solutions to the differential equation.

REMEMBER

But you don't have to know that solution to determine what the solutions behave like. You can tell just by looking at the direction field that all solutions tend toward a particular value, called the equilibrium value. For instance, you can see from the direction field graph in Figure 1-3 that the equilibrium value is 39.2. You also can see that equilibrium value in Figure 1-4.

Classifying Differential Equations

Tons of differential equations exist in Math and Science Land, and the way you tackle them differs by type. As a result, there are several classifications that you can put differential equations into. I explain them in the following sections.

Classifying equations by order

REMEMBER

The most common classification of differential equations is based on order. The order of a differential equation simply is the order of its highest derivative. For example, check out the following, which is a first order differential equation:

dy/dx = 5x

Here's an example of a second order differential equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

And so on, up to order n:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As you might imagine, first order differential equations are usually the most easily managed, followed by second order equations, and so on. I discuss first order, second order, and higher order differential equations in a bit more detail later in this chapter.

Classifying ordinary versus partial equations

You can also classify differential equations as ordinary or partial. This classification depends on whether you have only ordinary derivatives involved or only partial derivatives.

REMEMBER

An ordinary (non-partial) derivative is a full derivative, such as dQ/dt, where you take the derivative of all terms in Q with respect to t. Here's an example of an ordinary differential equation, relating the charge Q(t) in a circuit to the electromotive force E(t) (that is, the voltage source connected to the circuit):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, Q is the charge, L is the inductance of the circuit, ITLITL is the capacitance of the circuit, and E(t) is the electromotive force (voltage) applied to the circuit. This is an ordinary differential equation because only ordinary derivatives appear.

(Continues...)

Excerpted from Differential Equations For Dummiesby Steven Holzner Copyright © 2008 by Steven Holzner. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Edition details

• Publisher: For Dummies
• Publication date: 2008
• ISBN 10: 0470178140
• ISBN 13: 9780470178140
• Binding: Paperback
• Language: English
• Edition number: 1
• Number of pages: 368