Download presentation

Presentation is loading. Please wait.

Published byWilfrid Harris Modified over 6 years ago

1
First Order Linear Equations Integrating Factors

2
First Order Linear Equations Only First Derivative Of the form: Notice, we can rewrite:

3
First Order Linear Equations Only First Derivative Of the form: Notice, we can rewrite: So we consider general form:

4
Reversing Differentiation Rules Friday: Chain Rule Separation of Variables

5
Reversing Differentiation Rules Friday: Chain Rule Separation of Variables This is nonlinear

6
Reversing Differentiation Rules Friday: Chain Rule Separation of Variables We need a different technique.

7
Reversing Differentiation Rules Friday: Chain Rule Separation of Variables Today: Product Rule Integrating Factors

8
Product Rule

9
Want to rewrite as Then:

10
Product Rule Want to rewrite as Then:

11
Product Rule Want to rewrite as Then: Solution!

12
The Catch… is not of form

13
The Catch… (If it was, then )

14
So we introduce an Integrating Factor

18
(Challenge - Derive this using separation of variables)

19
Why? (Chain Rule) (Fundamental Theorem of Calculus)

20
So To Solve… Compute Integrating Factor Multiply Through Rewrit e Integrate

21
So To Solve… Compute Integrating Factor Multiply Through Rewrit e Integrate

22
Example Compute Integrating Factor

23
Example

24
Multiply Through By Integrating Factor

25
Example Multiply Through By Integrating Factor

26
Example Multiply Through By Integrating Factor

27
Example Rewrite Left Hand Side

28
Example Rewrite Left Hand Side

29
Example Integrate Both Sides

30
Example

31
(u sub)

32
Example Implicit Explicit

33
Long Term Behavior Solve s Equilibrium Solution:

34
Long Term Behavior Solve s Equilibrium Solution:

35
Long Term Behavior Solve s Equilibrium Solution:

36
Long Term Behavior Solve s Equilibrium Solution: What happens to general solutions over long times? For Any Constant C

37
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.

38
Long Term Behavior

39
Summary First Order Linear Equation: Solve Using Integrating Factors: Long term behavior determined by taking limit as t goes to infinity

40
Questions?

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google