separation of variables solving first order differential equations
TRANSCRIPT
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Separation of Variables
Solving First Order Differential Equations
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Solving ODEs
• What is Solving an ODE?
• Eliminating All Derivatives
Explicit Form
Implicit Form
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This Chapter
1st Order (Only First Derivative)
Linear and Nonlinear
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Calculus Brain Teaser:
?
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Calculus Brain Teaser:
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TodayWe will try to make problems look like:
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Why?Want to “Get Rid of”
This Derivative
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Why?
So we integrate the left side
Have to integrate right
side too
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Separation of Variables
No more derivatives! Implicit (General) Solution
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Separation of Variables
No more derivatives! Implicit (Specific) Solution
If we have can solve for C
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Chain Rule
Remember, y is a
function of t
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Chain Rule
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Chain Rule
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So To Solve
Think of it as:
(Reversing the Chain Rule)
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So To Solve
Think of it as:
Find by solving
Keep equation balanced by solving
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The whole process…For an equation of the
form:
(May need to manipulate equation to get here)
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The whole process…For an equation of the
form:
Separate the variables
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The whole process…For an equation of the
form:
Separate the variables
is is
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The whole process…For an equation of the
form:
Separate the variables
Integrate both sides
Perhaps solve for y, or C (if initial condition)
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A Simple Example
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A Simple Example
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A Simple Example
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A Simple Example
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A Simple Example
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A Simple Example
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A Simple Example
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A Convenient Technique
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A Convenient Technique
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A Convenient Technique
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A Convenient Technique
“Cross Multiply”
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A Convenient Technique
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A Convenient Technique
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A Convenient Technique
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Integral CurvesIs solved
by:
or
Equation for an ellipse (for different values of C)
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Integral Curves
Plots of Solutions for Different Values
of -C are called “Integral Curves”
Integral Curves Show Different Behaviors
for Different Initial Conditions
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Integral Curves
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Integral Curves
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Integral Curves
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In Summary
• To Solve an ODE, eliminate derivatives
• One method for first order linear/nonlinear ODES
• Separation of Variables (Reverse Chain Rule)
• Integral curves are solution curves for different values of C
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Questions?