Transcription of 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS
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1 INTRODUCTION TO DIFFERENTIAL EQUATIONS
11 INTRODUCTION TO and EQUATIONS as Mathematical ModelsCHAPTER 1 IN REVIEWThe words differentialandequationscertainly suggest solving some kind ofequation that contains derivatives y ,y , .. Analogous to a course in algebra andtrigonometry, in which a good amount of time is spent solving EQUATIONS such asx2 5x 4 0 for the unknown number x, in this course oneof our tasks will beto solve DIFFERENTIAL EQUATIONS such as y 2y y 0 for an unknown functiony (x).The preceding paragraph tells something, but not the complete story, about thecourse you are about to begin. As the course unfolds, you will see that there is moreto the study of DIFFERENTIAL EQUATIONS than just mastering methods that someone hasdevised to solve first things first. In order to read, study, and be conversant in a specializedsubject, you have to learn the terminology of that discipline. This is the thrust of thefirst two sections of this chapter.
First-order ordinary differential equations are occasionally written in differential form M(x, y) dx N(x, y) dy 0. For example, if we assume that y denotes the dependent variable in (y x) dx 4xdy 0, then y dy dx, so by dividing by the differential dx, we get the alternative form 4xy y x. See the Remarks at the end of this section. In symbols we ...
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