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

‘"Only skydivers know why the birds sing." From: http://www.adventureliving.com/home/skydiving/quotes/index.html Picture from: http://brisbane.jollypeople.com/files/2008/12/skydive.jpg http://www.youtube.com/watch?v=fq_fkHESF0c&feature=related

Plan for today Linear Systems

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).

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

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

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

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

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

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

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

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?

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

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

Group work

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.

The determinant

Is a solution of the system? Yes No What?

Is a solution of the system? Yes No What?

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.

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

Example

Verify the Linearity Principles

Example

Example

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.

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