Differential Equations – Day 2

Agenda:

  1. Finish Chapter 1
  2. Separable Equations
  3. Autonomous Equations

Generating Models

The need to describe a real world phenomenon!

  1. Identify the variables responsible for change in the system
  2. Make reasonable assumptions about the system
  3. Formulate a differential equation using these assumptions
  4. — Need to get the last two of these…

Model: Newton’s Cooling Law

The instantaneous rate of change of Temperature of an object is directly proportional to the difference between the temperatures of the object and the ambient temperature.

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Model: Spread of Disease

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The rate at which the disease spreads is proportional to the number of encounters/interactions between the infected and non infected groups. The number of interactions is jointly proportional to the two populations.

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In order to solve this, we need to introduce an infected individual and solve y(t) in terms of x and at time t there are n+1 individuals in the system.

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This was done to find a relationship between x and y to remove the y function above:

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Separable Equations

Any equation that can be written in the following form is considered separable.
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EX.

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And integrate…

Direction Fields & Autonomous Equations

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Phase Line

Equilibrium Solution, Constant Solution

Find the solutions (zeros) and divide your phase portrait into them. Use the first derivative test and try out values in those spectrum.

If they are convergent, it is an attractor. Stable

If they are divergent, it is a repeller. Unstable

If they are neither, it is semi-stable

Use second derivative test to check concavity & follow the path.

First order Linear ODE of the form, which is not separable:

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Find a function such that
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And multiply both sides by it, so that …
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By the product rule. And separate it to integrate!

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And
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And you end up getting the integrating factor. You use this and multiply it onto both sides to get the original ODE into a form that we can work with that is separable. =D