Über die Autorin bzw. den Autor

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.

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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

Aus dem Klappentext

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

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Differential Equations For Dummies

John Wiley & Sons

Chapter One

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...