NOTES FOR CALC III SHAUMS OUTLINE

1
NOTES FOR CALC IIISHAUMS OUTLINE

• CHAPTER 3
• SEPARABLE FIRST ORDER DIFFERENTIAL EQUATIONS

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Topics

• Separable
• Homogeneous

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Example 1

• Find the solution of the initial value problem
• Plot the graph of the solution
• Determine the interval in which the solution is
  defined

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Part a Solve
Separate the variables (in differential form)
Integrate both sides
Gives
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Part a Solve
From the last page
or
The condition gives
The solution is
or
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Part b Plot
3
-3
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Part c Interval of Definition
3
-3
From the graph
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Examples
2. Solve the initial value problem
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Examples
2. Solve the initial value problem
Separate the variables (in differential form)
Integrate both sides
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Solve
From the last page
gives
Using the initial condition
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The Graph of the Solution
We also know that
There is a horizontal tangent where this
derivative is zero
Or
gives
A graph of the solution
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Homogeneous Differential Equations
The goal is to substitute v y/x to change the
equation from the variables x and y to v and x
so that the equation becomes separable.
If v y/x then
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Steps for Solving Homogeneous Equations
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Example Solving Homogeneous Equations
Solve
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Example Solving Homogeneous Equations
Solve
Make the substitutions
Simplify
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Example Solving Homogeneous Equations
From the previous page
Subtract from both sides
Separate the variables
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Example Solving Homogeneous Equations
From the previous page

cut to the chase
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Example Solving Homogeneous Equations
Integrate and simplify
Re-substitute
Simplify
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END CHAPTER 3