First Order Linear Equations Integrating Factors.
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Transcript of First Order Linear Equations Integrating Factors.
First Order Linear Equations
Integrating Factors
First Order Linear Equations
Only First Derivative
Of the form:Notice, we can
rewrite:
First Order Linear Equations
Only First Derivative
Of the form:Notice, we can
rewrite:
So we consider general form:
Reversing Differentiation Rules
Friday:
Chain Rule
Separation of
Variables
Reversing Differentiation Rules
Friday:
Chain Rule
Separation of
Variables
This is nonlinear
Reversing Differentiation Rules
Friday:
Chain Rule
Separation of
Variables
We need a different technique.
Reversing Differentiation Rules
Friday:
Chain Rule
Separation of
VariablesToda
y:Product
Rule
Integrating
Factors
Product Rule
Product Rule
Want to rewrite as
Then:
Product Rule
Want to rewrite as
Then:
Product Rule
Want to rewrite as
Then:
Solution!
The Catch…
is not of form
The Catch…
(If it was, then )
So we introduce anIntegrating Factor
So we introduce anIntegrating Factor
So we introduce anIntegrating Factor
So we introduce anIntegrating Factor
So we introduce anIntegrating Factor
(Challenge - Derive this using separation of variables)
Why?
(Chain Rule)
(Fundamental Theorem of Calculus)
So To Solve…
Compute Integrating Factor
Multiply Through
Rewrite
Integrate
So To Solve…
Compute Integrating Factor
Multiply Through
Rewrite
Integrate
ExampleCompute Integrating
Factor
Example
Example
Multiply Through By Integrating Factor
Example
Multiply Through By Integrating Factor
Example
Multiply Through By Integrating Factor
Example
Rewrite Left Hand Side
Example
Rewrite Left Hand Side
Example
Integrate Both Sides
Example
Example
(u sub)
Example
Implicit
Explicit
Long Term BehaviorSolve
sEquilibrium Solution:
Long Term BehaviorSolve
sEquilibrium Solution:
Long Term BehaviorSolve
sEquilibrium Solution:
Long Term BehaviorSolve
sEquilibrium Solution:
What happens to general solutions over long times?
For Any Constant C
Long Term BehaviorSolve
sEquilibrium Solution:
What happens to general solutions over long times?
For Any Constant C
So we say solutions “converge” to y = 5/3
This also tells us that y = 5/3 is stable.
Long Term Behavior
Summary
• First Order Linear Equation:
• Solve Using Integrating Factors:
• Long term behavior determined by taking limit as t goes to infinity
Questions?