 # First Order Linear Equations Integrating Factors.

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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 Variables Today: Product Rule Integrating Factors

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 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

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

(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?