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You re ready to jump into 1 However, you don t have to start there if you don t want to; you can jump in anywhere you like this book was written to allow you to do just that But if you want to get the full story on differential equations from the beginning, jump into 1 first that s where all the action starts

n this part, I welcome you to the world of differential equations and start you off easy with linear first order differential equations With first order equations, you have first order derivatives that are raised to the first power, not squared or raised to any higher power I also show you how to work with separable first order differential equations, which are those equations that can be separated so that terms in y appear on one side and terms in x (and constants) appear on the other Finally, I introduce exact differential equations and Euler s method

1

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

t s a tense moment in the physics lab The international team of highpowered 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

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 1 2 3 4 5 6 Price $240 $250 $260 $270 $280 $290