Differential Equations

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Differential Equations

• Dillon Fadyn
• Spring 2000

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What is a differential equation?
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An equation with derivatives in it.
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Examples
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What was that last line?

• That is a partial differential equation (P.D.E.)
  because it has partial derivatives.

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Partial Derivatives?

• When u is a function of two variables, r and t,
  it has two partial first derivatives,
• one with respect to r,
• one with respect to t.

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Find the partial derivative of u with respect to
t by holding r constant and differentiating as
usual.
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Example

• Suppose

Then the partial derivative of u with respect to
t would be
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Notation
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Lingo
Second partial derivative of u with respect to t
Second mixed partial derivatives
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Dealing with Differential Equations

• Determine what the dependent variable is when you
  are presented with a differential equation.
• Determine what the independent variable(s) is
  (are), too.

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Example
Dependent variable
Independent variable
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Why?
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Example
Dependent variable
Independent variable
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Why?
as evidenced by the right hand side of the
equation.
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Example
Dependent variable
Independent variables
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Why?
as evidenced by the partial derivatives
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Why Does This Matter?

• We want solutions to differential equations.
• A solution to a differential equation is a
  function of the independent variable(s) which can
  successfully play the role of the dependent
  variable in the differential equation.

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In Other Words

• The unknown in a differential equation is the
  dependent variable.
• It is the thing we want to find.
• It is the thing whose derivatives appear in the
  differential equation.
• It is a function expressed in terms of the
  independent variable(s).

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Example
is a solution to
because
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Example
is a solution to
because
Check that this is true by calculating the
derivatives!
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Notice
In the last example
but we can call the independent variable
or anything we want.
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Exercise
Rewrite
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Example
is a solution to
because
Check that for homework!
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Homework

• Read Sections 1.1 and 1.2 in the text
• Answer the question, What is in the text that
  we didnt cover in class?
• On page 8, do problems 1-5, 10, 12, 13, 16-21
• On page 15, do problems 1, 2, 5, 11, 13, 16,
  18-22, 28-30, 35, 39