Overview
We now start our final topic for this course: Differential Equations. These are a special type of equations that involve derivatives, and they are especially useful in engineering and science. With differential equations, we can create models that explain the behavior of many real-world events.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Given a differential equation and a function \(f\), verify that \(f\) is a solution to the DE without attempting to solve the DE.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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State the meanings of differential equation (or DE), initial value problem, and autonomous differential equation in words.
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Recognize common situations in which a rate of change is given, and state how these are related to derivatives (such as interest rates, growth rates, and velocities).
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Given a common scenario describing a rate of change, translate this into a differential equation using mathematical notation.
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Recognize that there are many solutions to the same DE. Given a general solution, be able to find the particular solution satisfying a given initial condition.
To prepare for class
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Read all of Section 7.1 in Active Calculus.
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Watch the following video (by GVSUmath) which introduces the concept and use of differential equations:
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Watch the following video (by GVSUmath) which explains how exponential growth (for example a population growth) or decay (for example a radiactive decay) can be modelled by a differential equation:
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Watch the following video (by GVSUmath) which shows how to verify that a function is a solution of a differential equation:
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Do the Preview Activity for this section (on WeBWorK if required by your teacher).
