Powerpoint
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:
Friday:
Reversing Differentiation
Rules
Chain
Rule
Separation of
Variables
Friday:
Reversing Differentiation
Rules
Chain
Rule
Separation of
Variables
This is nonlinear
Friday:
Reversing Differentiation
Rules
Chain
Rule
Separation of
Variables
We need a different technique.
Friday:
Reversing Differentiation
Rules
Chain
Rule
Separation of
Variables
Today:
Product Rule
Integrating
Factors
Product Rule
Product Rule
Then:
Want to rewrite as
Product Rule
Then:
Want to rewrite as
Product Rule
Then:
Want to rewrite as
Solution!
The Catch… is not of form
The Catch…
(If it was, then )
So we introduce an
Integrating Factor
So we introduce an
Integrating Factor
So we introduce an
Integrating Factor
So we introduce an
Integrating Factor
So we introduce an
Integrating Factor
(Challenge - Derive this using separation of variables)
Why?
(Chain Rule)
(Fundamental Theorem of Calculus)
So To Solve…
Compute Integrating Factor
Multiply Through
Rewrit e
Integrate
So To Solve…
Compute Integrating Factor
Multiply Through
Rewrit e
Integrate
Example
Compute 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 Behavior
Solve s
Equilibrium Solution:
Long Term Behavior
Solve s
Equilibrium Solution:
Long Term Behavior
Solve s
Equilibrium Solution:
Long Term Behavior
Solve s
Equilibrium Solution:
What happens to general solutions over long times?
For Any Constant C
Long Term Behavior
Solve s
Equilibrium 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?
