Overview
We at long last see how to solve a differential equation - at least for a certain type of differential equation, the so-called “separable differential equations”. Along the way, we will learn how to use differential equations to construct some useful models for physical phenomena, solve them, and analyze their behavior.
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.
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Be able to put a given differential equation in the standard form \(\frac{dy}{dt} = g(y) h(t)\).
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Be able to solve simple separable differential equations.
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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Be able to solve a separable differential equation.
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Construct (from scratch) a differential equation that is a model for a given situation and solve it if it is a separable differential equation.
To prepare for class
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Watch the following video (by GVSUmath) which explains what makes a differential equation separable:
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Read the _Introduction to section 7.4 in _Active Calculus__.
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Do the Preview Activity for this section (on WeBWorK if required by your teacher).
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Read the detailed solution for Example 7.4.2 and pay careful attention to how the constant \(C\) is updated in the final steps of the solution. Note as well how the equilibrium solution is also taken into account in the final solution.
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Watch the following video (by patrickJMT) which shows another example of a separable differential equation. In particular, in this example, we are given an initial condition, and we obtain a single answer and not a family of solutions:
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Watch the following videos (by patrickJMT) which show more examples:
After class
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Watch the following videos which show examples of modeling real-life situations with differential equations (some of these are also described in section 7.5 of Active Calculus, which is optional reading):
