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 differentialequatio
  ns 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 (1)
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Examples of Fields Using Differential Equations
in Their Analysis (2)
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Examples of Fields Using Differential Equations
in Their Analysis (3)
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Examples of Fields Using Differential Equations
in Their Analysis (4)
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Examples of Fields Using Differential Equations
in Their Analysis (5)
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Differential Equation Basics

• The order of the highest derivative in a
  differential equation indicates the ORDER OF THE
  EQUATION.

• First Order Equation
• Second Order Equation
• Second Order Partial D.E.

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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
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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 variable
• 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