System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.
First Order Linear Equations Integrating Factors.
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Transcript of First Order Linear Equations Integrating Factors.
![Page 1: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/1.jpg)
First Order Linear Equations
Integrating Factors
![Page 2: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/2.jpg)
First Order Linear Equations
Only First Derivative
Of the form:Notice, we can
rewrite:
![Page 3: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/3.jpg)
First Order Linear Equations
Only First Derivative
Of the form:Notice, we can
rewrite:
So we consider general form:
![Page 4: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/4.jpg)
Reversing Differentiation Rules
Friday:
Chain Rule
Separation of
Variables
![Page 5: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/5.jpg)
Reversing Differentiation Rules
Friday:
Chain Rule
Separation of
Variables
This is nonlinear
![Page 6: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/6.jpg)
Reversing Differentiation Rules
Friday:
Chain Rule
Separation of
Variables
We need a different technique.
![Page 7: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/7.jpg)
Reversing Differentiation Rules
Friday:
Chain Rule
Separation of
VariablesToda
y:Product
Rule
Integrating
Factors
![Page 8: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/8.jpg)
Product Rule
![Page 9: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/9.jpg)
Product Rule
Want to rewrite as
Then:
![Page 10: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/10.jpg)
Product Rule
Want to rewrite as
Then:
![Page 11: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/11.jpg)
Product Rule
Want to rewrite as
Then:
Solution!
![Page 12: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/12.jpg)
The Catch…
is not of form
![Page 13: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/13.jpg)
The Catch…
(If it was, then )
![Page 14: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/14.jpg)
So we introduce anIntegrating Factor
![Page 15: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/15.jpg)
So we introduce anIntegrating Factor
![Page 16: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/16.jpg)
So we introduce anIntegrating Factor
![Page 17: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/17.jpg)
So we introduce anIntegrating Factor
![Page 18: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/18.jpg)
So we introduce anIntegrating Factor
(Challenge - Derive this using separation of variables)
![Page 19: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/19.jpg)
Why?
(Chain Rule)
(Fundamental Theorem of Calculus)
![Page 20: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/20.jpg)
So To Solve…
Compute Integrating Factor
Multiply Through
Rewrite
Integrate
![Page 21: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/21.jpg)
So To Solve…
Compute Integrating Factor
Multiply Through
Rewrite
Integrate
![Page 22: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/22.jpg)
ExampleCompute Integrating
Factor
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Example
![Page 24: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/24.jpg)
Example
Multiply Through By Integrating Factor
![Page 25: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/25.jpg)
Example
Multiply Through By Integrating Factor
![Page 26: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/26.jpg)
Example
Multiply Through By Integrating Factor
![Page 27: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/27.jpg)
Example
Rewrite Left Hand Side
![Page 28: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/28.jpg)
Example
Rewrite Left Hand Side
![Page 29: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/29.jpg)
Example
Integrate Both Sides
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Example
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Example
(u sub)
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Example
Implicit
Explicit
![Page 33: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/33.jpg)
Long Term BehaviorSolve
sEquilibrium Solution:
![Page 34: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/34.jpg)
Long Term BehaviorSolve
sEquilibrium Solution:
![Page 35: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/35.jpg)
Long Term BehaviorSolve
sEquilibrium Solution:
![Page 36: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/36.jpg)
Long Term BehaviorSolve
sEquilibrium Solution:
What happens to general solutions over long times?
For Any Constant C
![Page 37: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/37.jpg)
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.
![Page 38: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/38.jpg)
Long Term Behavior
![Page 39: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/39.jpg)
Summary
• First Order Linear Equation:
• Solve Using Integrating Factors:
• Long term behavior determined by taking limit as t goes to infinity
![Page 40: First Order Linear Equations Integrating Factors.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649dbc5503460f94aadff1/html5/thumbnails/40.jpg)
Questions?