Lecture 25: Ordinary Differential Equations 1 of 2

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Lecture (25) Ordinary Differential Equations (1
of 2)

• A differential equation is an algebraic equation
  that contains some derivatives

• Recall that a derivative indicates a change in a
  dependent variable with respect to an independent
  variable.
• In these two examples, y is the dependent
  variable and t and x are the independent
  variables, respectively.

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Why study differential equations?

• Many descriptions of natural phenomena are
  relationships (equations) involving the rates at
  which things happen (derivatives).
• Equations containing derivatives are called
  differential equations.
• Ergo, to investigate problems in many fields of
  science and technology, we need to know something
  about differential equations.

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Why study differential equations?

• Some examples of fields using differentialequati
  ons in their analysis include

• Solid mechanics motion
• heat transfer energy balances
• vibrational dynamics seismology
• aerodynamics fluid dynamics
• electronics circuit design
• population dynamics biological systems
• climatology and environmental analysis
• options trading economics

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Examples of Fields Using Differential Equations
in Their Analysis
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Differential Equation Basics

• The order of the highest derivative ina
  differential equation indicates the order of the
  equation.

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Simple Differential Equations
A simple differential equation has the form
Its general solution is
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Simple Differential Equations
Ex. Find the general solution to
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Simple Differential Equations
Ex. Find the general solution to
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Exercise (Waner, Problem 1, Section 7.6)
Find the general solution to
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Example Motion
A drag racer accelerates from a stop so that its
speed is 40t feet per second t seconds after
starting. How far will the car go in 8 seconds?
Given
Find
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Solution
Apply the initial condition s(0) 0
The car travels 1280 feet in 8 seconds
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Exercise (Waner, Problem 11, Section 7.6)
Find the particular solution to
Apply the initial condition y(0) 1
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Separable Differential Equations
A separable differential equation has the form
Its general solution is
Example Separable Differential Equation
Consider the differential equation
a. Find the general solution. b. Find the
particular solution that satisfies the initial
condition y(0) 2.
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Solution
a.
Step 1 Separate the variables
Step 2 Integrate both sides
Step 3 Solve for the dependent variable
This is the general solution
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Solution (continued)
b.
Apply the initial (or boundary) condition, that
is, substituting 0 for x and 2 for y into the
general solution in this case, we get
Thus, the particular solution we are looking for
is
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Exercise (Waner, Problem 4, Section 7.6)
Find the general solution to