1. Applied Differential Equations I
Sebastian M. Marotta
Department of Mathematics
University of the Pacific
Lecture 18
October 16, 2009

2. ‘"Only skydivers know why the birds sing."
From:
Picture from:

3. Plan for today
Linear Systems

4. Summary of the course content in one slide:
Modeling.
First order differential equations.
Qualitative, Analytic and Numerical Techniques.
Autonomous Vs. Non-autonomous equations.
Bifurcations.
Linear Equations (lucky guess and integrating factors).
Linear, homogeneous, non-homogeneous equations.
First order systems (second order differential equations).
Linear systems
The mass-spring system with forcing (resonance).
Nonlinear systems.
Laplace transforms (analytic technique).

5. Homework 12 (to do by Wednesday)
Section 3.1: # 4, 6, 9, 11, 12, 16, 24, 28, 33, 34.

6. Linear Systems
a, b, c and d are constants.
Matrix notation

7. Is this a linear system?
Yes No
I don’t know

8. Is this a linear system?
Yes No
I don’t know

9. A mass-spring system Wall Spring Mass: m Surface
Second Order Linear Ordinary Differential Equation with Constant Coefficients.

10. A mass-spring system Wall Spring Mass: m Surface y < 0 y > 0

11. First Order Linear System
A mass-spring system
Wall
Spring
Mass: m
Surface
y < 0
y > 0
y = 0
First Order Linear System

12. What is the matrix associated to this linear system?
A mass-spring system
Wall
Spring
Mass: m
Surface
y < 0
y > 0
y = 0
What is the matrix associated to this linear system?

13. What is the matrix associated to the mass-spring linear system?

14. Linear Systems: the determinant
a, b, c and d are constants.
Matrix notation

15. Group work

16. Exercise
1. Find the equilibrium solutions of each of the following systems.
2. Compute the determinant of the associated matrix.
3. Compare the data found in 1 and 2. What do you conclude?
Work on the boards.

17. The determinant

18. Is a solution of the system?
Yes No
What?

19. Is a solution of the system?
Yes No
What?

20. Linearity Principles
If Y(t) is a solution then kY(t) is also a solution.
If Y1(t) and Y2(t) are two solutions then Y1(t) + Y2(t) is also a solution.
Then any linear combination k1Y1(t) + k2Y2(t) is also a solution for any constants k1 and k2.

21. Example
Exercise: Draw the three solutions in the phase plane.

22. Example

23. Verify the Linearity Principles

24. Example

25. Example

26. The General Solution
If Y1(t) and Y2(t) are two linearly independent solutions then the general solution of the system is given by
Y(t) = k1Y1(t) + k2Y2(t) for constants k1 and k2.

27. For Monday Study Section 3.2
What is the speed of a sky diver 5 seconds after the jump?