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Week 5 Feb 15-19

by: Susan Ossareh

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Week 5 Feb 15-19 Math 340

Susan Ossareh
CSU

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These notes cover topics 2.9 and 6.1 with a lot of examples
COURSE
Intro-Ordinary Differen Equatn
PROF.
TYPE
Class Notes
PAGES
4
WORDS
CONCEPTS
Math, Differential Equations
KARMA
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This 4 page Class Notes was uploaded by Susan Ossareh on Sunday February 21, 2016. The Class Notes belongs to Math 340 at Colorado State University taught by in Spring 2016. Since its upload, it has received 23 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

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Date Created: 02/21/16
Math 340 Lecture – Introduction to Ordinary Differential Equations – February 15, 2016 What We Covered: 1. Worksheet 05: Equilibrium Solutions and Stability a. Highlights i. We worked through the worksheet the entire time, make sure you understand the content because our quiz will cover this topic ii. The worksheet questions were as follows: 1. Consider the differential equation ???? = 1 − ???? and its direction field showing some of the solutions a. Explain why there is a different solution for every initial value (????0,????0) b. Using the information from the direction field, describe the behavior as ???? → ∞ of the solution with the given initial condition (????0,????0) i. ???? 0 = 1: ii. ???? 0 = −1: iii. −1 < ???? 0 < 1: iv. ???? 0 > 1: v. ???? 0 < −1: c. Identify the equilibrium solutions d. Classify the equilibrium solutions as unstable of asymptotically stable ′ 3 2 2. Find and classify the equilibrium solutions of ???? = −???? − ???? + 2???? + 2. Draw the phase line and 5 different solutions on the tx-plane ′ ( ) 3. Find and classify the equilibrium solutions of ???? = 1 − ???? ln⁡(1 + ????). Draw the phase line and 5 different solutions on the tx-plane. 4. Consider the logistic model for population growth given by ???? = ′ ???? ???? (1 − )???? where r is the natural reproductive rate of the population and ???? k is the carrying capacity a. Find the equilibrium solutions for the equation and the conditions that make them unstable of asymptotically stable b. Find the general solution Suggested Homework:  Finish worksheet 5  Study for the quiz this Wednesday Math 340 Lab – Introduction to Ordinary Differential Equations – February 16, 2016 What We Covered: 1. Course Content – Chapter 2: First Order Equations a. Section 2.9: Autonomous Equations and Stability Continued i. Let’s go through worksheet 5, question 2 step by step and break down what we’ve been learning 1. So suppose we are given ???? = −???? − ???? + 2???? + 2. We say ???? = ????(????) a. Step one would be to set f(x) equal to zero because we want to find the equilibrium points. ???? ???? = 0 −???? − ???? + 2???? + 2 = 0 2 −???? (???? + 1 + 2 ???? + 1 = 0 (???? + 1 2 − ???? 2) = 0 (???? + 1 (√2 − ????)( √ + ????) = 0 ???? = 1⁡⁡⁡⁡⁡???? =2⁡⁡⁡⁡⁡???? = − 2 √ √ b. We can then set them up to find the equilibrium solutions, which are essentially the same as the points but we need to format it into an equation ???? ???? = −1⁡⁡⁡⁡⁡???? ???? = √ 2⁡⁡⁡⁡⁡???? ???? = √ 2 c. From here, we can draw a phase line. A phase line describes the dynamics involved in the motion of y(t) along the line. We can see how the function would ovulate graphically which can tell us if the equilibrium solutions are stable or unstable i. To check for stability, you use the first derivative test. If f’(x) is negative, then the arrow points left on the phase line and is asymptotically stable. If f’(x) is positive, the arrow points right on the phase line and is unstable. If f’(x) is 0, it’s inconclusive ii. So let’s solve the equilibrium points to see if they are stable or unstable! ′ 3 2 ???? = −???? − ???? + 2???? + 2 ???? ???? = −3???? − 2???? + 2 x(t)=-1 ???? −1 = −3 + 2 + 2 = 1 1 > 0⁡????ℎ????????????????????????????,???? ???? = −1⁡????????⁡???????????????????????????????? x(t)=√2 ???? (√2) = −3( √) − 2( 2)√+ 2 = −4 − 2 2 < √ ???? ???? = √ 2⁡????????⁡????????????⁡???????????????????????? x(t)=-√2 ′ 2 ???? (−√2) = −3(− 2)√− 2(− 2) +√2 = −4 + 2 2√< 0,????ℎ????????????????????????????⁡???? ???? = − 2⁡????????√????????????⁡???????????????????????? Suggested Homework:  Section 2.9: 5, 7, 10, 11, 17, 22 Math 340 Lecture – Introduction to Ordinary Differential Equations – February 17, 2016 What We Covered: 1. Quiz a. Highlights i. Covers the topics Existence and Uniqueness and Equilibrium Solutions 2. Course Content – Chapter 2: First Order Equations a. Section 2.9: Autonomous Equations and Stability Continued i. Summary of Method: So let’s go over how to analyze the solutions of an autonomous equation 1. Graph the right-hand side f(x) and add the phase line information to the x-axis. Find the equilibrium points which is where f(x)=0. In each intervals limited by the equilibrium points, find the sign of f and draw an arrow t the right if f is positive and to the left if f is negative 2. Create a tx-plane and transfer the information from the phase line, to the new graph. Sketch the nonequilibrium solutions in each interval limited by the equilibrium points ii. Example: Find and classify the equilibrium points and solutions of ???? = ???? −′ 3 2 2???? + ????. Always make sure everything on the right hand side is x ???? ???? = 0 3 2 ???? − 2???? + ???? = 0 ???? ???? − 2???? + 1 = 0 2 ????(???? − 1) = 0 ???? = 0⁡⁡⁡⁡⁡???? = 1 1. Now format these points into the equilibrium solutions ???? ???? = ???? − 2???? + ???? 2 ′ 2 ???? ???? = 3???? − 4???? + 1 2. Okay, so the next step is to create the phase line, we do this by the first derivative test which tells us whether the solutions are stable, unstable, or inconclusive x(t)=0 ???? 0 = 3 0 − 4 0 + 1 = 1 1 > 0,????ℎ????????????????????????????⁡???? ???? = 0⁡????????⁡???????????????????????????????? x(t)=1 ′ ???? 1 = 3 1 − 4 1 + 1 = 0 0 = 0,????ℎ????????????????????????????⁡???? ???? = 0⁡????????⁡???????????????????????????????????????????????? iii. Existence and Uniqueness Review 1. Does the existence theorem guarantee a solution and for which initial value? a. F is continuous everywhere so there is a solution 2. How many solutions are there from each (???? ,???? ) 0 0 a. F(x) is continuous ???????? b. ???????? is continuous c. There is only one solution by the uniqueness theorem 3. Course Content – Chapter 6: Numerical Methods a. Section 6.1: Euler’s Method i. Euler’s method is an example of a fixed-step solver which means there are a discrete set of values of the independent variable so there the interval is divided into equal subintervals to approximate the solution ii. Let’s walk through the general idea and next class we can actually do some examples to make sense of this topic ′ ( ) ( ) 1. Given ???? = ???? ????,???? ⁡⁡⁡⁡→ ⁡⁡⁡⁡⁡???? ???? 0 = ???? 0 2. We know that f(t,y) gives the slope of y(t) at every point 3. So ????(???? ,???? ) is the slope of y(t) at (???? ,???? ) 0 0 0 0 4. Remember that the equation of the line is given by a line that contains (????0,???? 0: ???? = ???? 0 ????(???? − ???? ) 0 ????(????0,????0) 5. So the tangent line to y(t) at (???? ,???? ) is ???? ???? = ???? + ????(???? ,???? )(???? − ???? ) 0 0 0 0 0 0 ????1= ???? ????(1) = ???? 0 ????(???? ,0 )(0 − ???? ) 0 ???? = ???? + ???? ???? ,???? ℎ ) 1 0 0 0 In general: ???? = ???? + ???? ???? ,???? )ℎ ???? ????−1 ????−1 ????−1 Suggest Homework:  Section 2.9: 5, 7, 10, 11, 17, 22  Section 6.1: 4, 6, 8 Math 340 Lecture – Introduction to Ordinary Differential Equations – February 19, 2016 What We Covered: 1. Worksheet 6: Euler’s Method a. Highlights i. Compute the first 3 steps of Euler’s method to approximate the solution of ???? = ′ ???? − ???? with ???? 1 = 1. Use ℎ = 0.1. ′ ii. Compute the first 4 steps of Euler’s method to approximate the solution of ???? = 2???? − 1 with ???? 0 = 0. Partition the interval [0,2]. ′ iii. Compute the first 4 steps of Euler’s method to approximate the solution of ???? = 2???? − 1 with ???? 0 = 0. Use ℎ = 0.25 ′ iv. Solve the equation ???? = 2???? − 1 with ???? 0 = 0 and sketch the solution curve you obtained using the values of ????:0,0.5,1,1.5,2. Sketch the approximation from the last 2 examples in the same plane. Which of the 2 approximations from the last 2 examples in the same plane. Which of the 2 approximations is closer to the solution? What would you do to get an even better approximation? b. To solve problems 1 through 3… i. Partition the interval [a,b] ????−???? ii. Find h=length of each step where ℎ = ???? iii. Compute ???? (????????) ????hich is an approximate iv. ???? = ???? + ???? ???? ,???? )ℎ where ℎ = (???? − ???? ) ???? ????−1 ????−1 ????−1 ???? ????−1 Suggested Homework:  Section 6.1: 4, 6, 8  Finish worksheet 6

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