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Published byWilfrid Harris Modified over 4 years ago

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First Order Linear Equations Integrating Factors

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

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First Order Linear Equations Only First Derivative Of the form: Notice, we can rewrite: So we consider general form:

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Reversing Differentiation Rules Friday: Chain Rule Separation of Variables

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Reversing Differentiation Rules Friday: Chain Rule Separation of Variables This is nonlinear

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Reversing Differentiation Rules Friday: Chain Rule Separation of Variables We need a different technique.

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Reversing Differentiation Rules Friday: Chain Rule Separation of Variables Today: Product Rule Integrating Factors

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Product Rule

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Want to rewrite as Then:

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Product Rule Want to rewrite as Then:

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Product Rule Want to rewrite as Then: Solution!

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The Catch… is not of form

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The Catch… (If it was, then )

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So we introduce an Integrating Factor

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(Challenge - Derive this using separation of variables)

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Why? (Chain Rule) (Fundamental Theorem of Calculus)

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So To Solve… Compute Integrating Factor Multiply Through Rewrit e Integrate

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So To Solve… Compute Integrating Factor Multiply Through Rewrit e Integrate

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Example Compute Integrating Factor

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Example

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Multiply Through By Integrating Factor

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Example Multiply Through By Integrating Factor

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Example Multiply Through By Integrating Factor

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Example Rewrite Left Hand Side

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Example Rewrite Left Hand Side

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Example Integrate Both Sides

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Example

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(u sub)

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Example Implicit Explicit

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Long Term Behavior Solve s Equilibrium Solution:

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Long Term Behavior Solve s Equilibrium Solution:

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Long Term Behavior Solve s Equilibrium Solution:

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Long Term Behavior Solve s Equilibrium Solution: What happens to general solutions over long times? For Any Constant C

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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.

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Long Term Behavior

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Summary First Order Linear Equation: Solve Using Integrating Factors: Long term behavior determined by taking limit as t goes to infinity

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Questions?

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