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?