Introduction to Differential Equations

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Introduction to Differential Equations

• MATH 224

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Textbook Reminder

• In the text, the second half is "Differential
  Equations" Ziff
• Section numbers now refer to this part of the text

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Why???
Model
Physical System
Check
Prediction of Behaviour
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Example 1 Heating and Cooling

• System air in unheated building
• Assumptions

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Rates of Change

• Model

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What do you do next?

• You now have a differential equation!
• You are looking for _______________
• Mathematically, this is

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Graphs

• DE was

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Properties of Solutions

• General Solution
• Particular Solution

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Finding the General Solution

• To find the function f(t) for the temperature
  over time, we can take an educated guess for this
  DE

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Finding the General Solution
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Finding a Particular Solution

• Say the current temperature in the house is 15
  degrees. What will the temperature be as a
  function of time?

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Qualitative Analysis I

• Why is -5 a constant solution?

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Qualitative analysis II

• Why does temperature drop when above -5?
• Why does it grow when below -5?

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Check

• Are predictions reasonable?
• What changes could be made to improve accuracy of
  predictions?

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Question
Which of the following shows solutions to
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Question
Which of the following shows solutions to
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Some Notes

• Differential equation is the mathematical form of
  a physical law
• Many physical laws naturally expressed as DEs
• Law covers cases we might think are different
• cooling vs heating

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Other Applications of DEs

• Any system where model can be expressed using
• rates of change
• slopes, curvature

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DEs in Civil Engineering

• beam bending (statics)
• water flow (dams, rivers, tides, waves)
• column buckling
• contaminant flows (groundwater)
• seepage
• dynamics of structures

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Reading

• Sections 1.1 and 1.3 in DE text,
• More context if you read Project 1 (pg 36)