First Order Linear
Equations
Integrating Factors

Only First
Derivative
Of the form:
Notice, we can rewrite:

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?