Ordinary Differential Equations (ODEs) Background. Video 17

Size: px
Start display at page:

Download "Ordinary Differential Equations (ODEs) Background. Video 17"

Transcription

1 Ordinary Differential Equations (ODEs) Background Video 17 Daniel J. Bodony Department of Aerospace Engineering University of Illinois at Urbana-Champaign

2 In this video you will learn... 1 What ODEs are 2 Matlab and ODEs: first order form D. J. Bodony (UIUC) AE199 IAC Video 17 2 / 9

3 What ODEs Are Let s introduce ODEs by way of a familiar example: a ballistic trajectory. Consider this picture of a sphere of mass m leaving a planar surface with initial speed v 0 and an angle θ 0 from the x-axis: How do you find the trajectory? You apply Newton s 2nd Law F = m a on the sphere to get ma x = 0 ma y = mg. D. J. Bodony (UIUC) AE199 IAC Video 17 3 / 9

4 What ODEs Are From the definition of acceleration we know that the velocity v and acceleration are related via d v dt = a and the position x(t) of the sphere is also given by Putting all of these together yields d x dt = v. m d2 x dt 2 = 0 m d2 y dt 2 = mg. D. J. Bodony (UIUC) AE199 IAC Video 17 4 / 9

5 What ODEs Are Assuming g is a constant, we can integrate these two equations to get x(t) = x 0 + v 0 t cos θ 0 y(t) = y 0 + v 0 t sin θ 0 gt2 2 for the trajectory x(t) = (x(t), y(t)) of the sphere. D. J. Bodony (UIUC) AE199 IAC Video 17 5 / 9

6 What ODEs Are Assuming g is a constant, we can integrate these two equations to get x(t) = x 0 + v 0 t cos θ 0 y(t) = y 0 + v 0 t sin θ 0 gt2 2 for the trajectory x(t) = (x(t), y(t)) of the sphere. So what? Well, go back to the equations m d2 x dt 2 = 0 m d2 y dt 2 = mg from the previous slide. Notice they are of the form {some derivative of at least one function} = {another function} This is precisely what we mean by an ordinary differential equation. D. J. Bodony (UIUC) AE199 IAC Video 17 5 / 9

7 What ODEs Are Going back to Newton s 2nd Law, we are free to write it as (for a constant mass m) d 2 x dt 2 = F m as an ODE for the trajectory x(t) as a function of the applied forces F and mass m. D. J. Bodony (UIUC) AE199 IAC Video 17 6 / 9

8 What ODEs Are Going back to Newton s 2nd Law, we are free to write it as (for a constant mass m) d 2 x dt 2 = F m as an ODE for the trajectory x(t) as a function of the applied forces F and mass m. When we found the solution for the trajectory of the sphere of mass m, we used two other ODEs d v dt = a d x dt = v. So, in reality, you already know a lot about ODEs. D. J. Bodony (UIUC) AE199 IAC Video 17 6 / 9

9 Matlab and ODEs: first order form Matlab has several functions that integrate ODEs for you, even the ones you can t do by hand. There s a catch, though: you have to write the ODEs in first order form which means that the equations must look like this {first derivative of one function} = {another function}. In terms of our trajectory example this means the ODE must be written as instead of as d v dt = F m d x = v. }{{} dt first deriv. d 2 x F }{{} dt 2 = m. second deriv. D. J. Bodony (UIUC) AE199 IAC Video 17 7 / 9

10 Matlab and ODEs: first order form Matlab generically refers to the ODE as d y dt = y = f (t, y) where f is known as the right-hand-side function. 1 You ll learn much more about it in AE370. D. J. Bodony (UIUC) AE199 IAC Video 17 8 / 9

11 Matlab and ODEs: first order form Matlab generically refers to the ODE as d y dt = y = f (t, y) where f is known as the right-hand-side function. The most heavily used ODE integrator in Matlab is ode45. There is lots of details about what ode45 is, how it works, when it fails, etc.; however, the only you thing you need to know is how to use it. 1 1 You ll learn much more about it in AE370. D. J. Bodony (UIUC) AE199 IAC Video 17 8 / 9

12 Matlab and ODEs: first order form The ode45 function call looks like this: where [time, out] = ode45(@fun, tspan, ic); time tspan ic = column vector of time = matrix of the solution (more later on this) = function handle of f (t, y) = row vector of [t 0, t f ], the initial and final times = row vector of initial conditions D. J. Bodony (UIUC) AE199 IAC Video 17 9 / 9

13 Matlab and ODEs: first order form The ode45 function call looks like this: where [time, out] = ode45(@fun, tspan, ic); time tspan ic = column vector of time = matrix of the solution (more later on this) = function handle of f (t, y) = row vector of [t 0, t f ], the initial and final times = row vector of initial conditions In the next video we ll do some examples. D. J. Bodony (UIUC) AE199 IAC Video 17 9 / 9

Chapter 9b: Numerical Methods for Calculus and Differential Equations. Initial-Value Problems Euler Method Time-Step Independence MATLAB ODE Solvers

Chapter 9b: Numerical Methods for Calculus and Differential Equations. Initial-Value Problems Euler Method Time-Step Independence MATLAB ODE Solvers Chapter 9b: Numerical Methods for Calculus and Differential Equations Initial-Value Problems Euler Method Time-Step Independence MATLAB ODE Solvers Acceleration Initial-Value Problems Consider a skydiver

More information

Course roadmap. ME451: Control Systems. Example of Laplace transform. Lecture 2 Laplace transform. Laplace transform

Course roadmap. ME451: Control Systems. Example of Laplace transform. Lecture 2 Laplace transform. Laplace transform ME45: Control Systems Lecture 2 Prof. Jongeun Choi Department of Mechanical Engineering Michigan State University Modeling Transfer function Models for systems electrical mechanical electromechanical Block

More information

Computational Project 5

Computational Project 5 Last updated December 5, 2017. Computational Project 5 due: 12/07/2017, beginning of class 1. You are playing a game of angry birds. In order to hit the pig you need to first shoot your bird through a

More information

Initial value problems for ordinary differential equations

Initial value problems for ordinary differential equations AMSC/CMSC 660 Scientific Computing I Fall 2008 UNIT 5: Numerical Solution of Ordinary Differential Equations Part 1 Dianne P. O Leary c 2008 The Plan Initial value problems (ivps) for ordinary differential

More information

Solving systems of ODEs with Matlab

Solving systems of ODEs with Matlab Solving systems of ODEs with Matlab James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 20, 2013 Outline 1 Systems of ODEs 2 Setting Up

More information

Mathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Spring Department of Mathematics

Mathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Spring Department of Mathematics Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs)

More information

Mathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics

Mathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs) is

More information

Section 8.0 Introduction to Boundary Value Problems. How do initial value problems (IVPs) and boundary value problems (BVPs) differ?

Section 8.0 Introduction to Boundary Value Problems. How do initial value problems (IVPs) and boundary value problems (BVPs) differ? Section 8.0 Introduction to Boundary Value Problems Key terms/ideas How do initial value problems (IVPs) and boundary value problems (BVPs) differ? What are boundary conditions? In what type of problems

More information

Predicting the future with Newton s Second Law

Predicting the future with Newton s Second Law Predicting the future with Newton s Second Law To represent the motion of an object (ignoring rotations for now), we need three functions x(t), y(t), and z(t), which describe the spatial coordinates of

More information

Physics 584 Computational Methods

Physics 584 Computational Methods Physics 584 Computational Methods Introduction to Matlab and Numerical Solutions to Ordinary Differential Equations Ryan Ogliore April 18 th, 2016 Lecture Outline Introduction to Matlab Numerical Solutions

More information

Calculus of Variations Summer Term 2015

Calculus of Variations Summer Term 2015 Calculus of Variations Summer Term 2015 Lecture 14 Universität des Saarlandes 24. Juni 2015 c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 1 / 20 Purpose of Lesson Purpose

More information

The integrating factor method (Sect. 1.1)

The integrating factor method (Sect. 1.1) The integrating factor method (Sect. 1.1) Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Overview

More information

Scientific Computing with Case Studies SIAM Press, Lecture Notes for Unit V Solution of

Scientific Computing with Case Studies SIAM Press, Lecture Notes for Unit V Solution of Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit V Solution of Differential Equations Part 1 Dianne P. O Leary c 2008 1 The

More information

Differentiation of Parametric Space Curves. Goals: Velocity in parametric curves Acceleration in parametric curves

Differentiation of Parametric Space Curves. Goals: Velocity in parametric curves Acceleration in parametric curves Block #2: Differentiation of Parametric Space Curves Goals: Velocity in parametric curves Acceleration in parametric curves 1 Displacement in Parametric Curves - 1 Displacement in Parametric Curves Using

More information

Volumes of Solids of Revolution Lecture #6 a

Volumes of Solids of Revolution Lecture #6 a Volumes of Solids of Revolution Lecture #6 a Sphereoid Parabaloid Hyperboloid Whateveroid Volumes Calculating 3-D Space an Object Occupies Take a cross-sectional slice. Compute the area of the slice. Multiply

More information

Lecture 5. Dynamics. Forces: Newton s First and Second

Lecture 5. Dynamics. Forces: Newton s First and Second Lecture 5 Dynamics. Forces: Newton s First and Second What is a force? It s a pull or a push: F F Force is a quantitative description of the interaction between two physical bodies that causes them to

More information

ACM/CMS 107 Linear Analysis & Applications Fall 2016 Assignment 4: Linear ODEs and Control Theory Due: 5th December 2016

ACM/CMS 107 Linear Analysis & Applications Fall 2016 Assignment 4: Linear ODEs and Control Theory Due: 5th December 2016 ACM/CMS 17 Linear Analysis & Applications Fall 216 Assignment 4: Linear ODEs and Control Theory Due: 5th December 216 Introduction Systems of ordinary differential equations (ODEs) can be used to describe

More information

Chapter 3: Second Order ODE 3.8 Elements of Particle Dy

Chapter 3: Second Order ODE 3.8 Elements of Particle Dy Chapter 3: Second Order ODE 3.8 Elements of Particle Dynamics 3 March 2018 Objective The objective of this section is to explain that any second degree linear ODE represents the motion of a particle. This

More information

Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories

Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 2013 Outline

More information

There seems to be three different groups of students: A group around 6 A group around 12 A group around 16

There seems to be three different groups of students: A group around 6 A group around 12 A group around 16 10 5 0 0 5 10 15 20 25 30 There seems to be three different groups of students: A group around 6 A group around 12 A group around 16 Altuğ Özpineci ( METU ) Phys109-MECHANICS PHYS109 55 / 67 10 5 0 0 5

More information

Lecture 6, September 1, 2017

Lecture 6, September 1, 2017 Engineering Mathematics Fall 07 Lecture 6, September, 07 Escape Velocity Suppose we have a planet (or any large near to spherical heavenly body) of radius R and acceleration of gravity at the surface of

More information

Lecture 2. Introduction to Differential Equations. Roman Kitsela. October 1, Roman Kitsela Lecture 2 October 1, / 25

Lecture 2. Introduction to Differential Equations. Roman Kitsela. October 1, Roman Kitsela Lecture 2 October 1, / 25 Lecture 2 Introduction to Differential Equations Roman Kitsela October 1, 2018 Roman Kitsela Lecture 2 October 1, 2018 1 / 25 Quick announcements URL for the class website: http://www.math.ucsd.edu/~rkitsela/20d/

More information

34.3. Resisted Motion. Introduction. Prerequisites. Learning Outcomes

34.3. Resisted Motion. Introduction. Prerequisites. Learning Outcomes Resisted Motion 34.3 Introduction This Section returns to the simple models of projectiles considered in Section 34.1. It explores the magnitude of air resistance effects and the effects of including simple

More information

Differential Equation (DE): An equation relating an unknown function and one or more of its derivatives.

Differential Equation (DE): An equation relating an unknown function and one or more of its derivatives. Lexicon Differential Equation (DE): An equation relating an unknown function and one or more of its derivatives. Ordinary Differential Equation (ODE): A differential equation that contains only ordinary

More information

MA 102 Mathematics II Lecture Feb, 2015

MA 102 Mathematics II Lecture Feb, 2015 MA 102 Mathematics II Lecture 1 20 Feb, 2015 Differential Equations An equation containing derivatives is called a differential equation. The origin of differential equations Many of the laws of nature

More information

Matlab Course. Anna Kristine Wåhlin. Department of Geophysics, University of Oslo. January Matlab Course p.1/??

Matlab Course. Anna Kristine Wåhlin. Department of Geophysics, University of Oslo. January Matlab Course p.1/?? Matlab Course Anna Kristine Wåhlin Department of Geophysics, University of Oslo January 2003 Matlab Course p.1/?? Numerical estimate of the derivative An estimate of the time derivative of dataset at time

More information

Physics 202 Laboratory 3. Root-Finding 1. Laboratory 3. Physics 202 Laboratory

Physics 202 Laboratory 3. Root-Finding 1. Laboratory 3. Physics 202 Laboratory Physics 202 Laboratory 3 Root-Finding 1 Laboratory 3 Physics 202 Laboratory The fundamental question answered by this week s lab work will be: Given a function F (x), find some/all of the values {x i }

More information

Calculus of Variations Summer Term 2016

Calculus of Variations Summer Term 2016 Calculus of Variations Summer Term 2016 Lecture 14 Universität des Saarlandes 28. Juni 2016 c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 28. Juni 2016 1 / 31 Purpose of Lesson Purpose

More information

Computational Neuroscience. Session 1-2

Computational Neuroscience. Session 1-2 Computational Neuroscience. Session 1-2 Dr. Marco A Roque Sol 05/29/2018 Definitions Differential Equations A differential equation is any equation which contains derivatives, either ordinary or partial

More information

MATH 100 Introduction to the Profession

MATH 100 Introduction to the Profession MATH 100 Introduction to the Profession Differential Equations in MATLAB Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2012 fasshauer@iit.edu MATH 100 ITP 1 What

More information

Lecture 9. Systems of Two First Order Linear ODEs

Lecture 9. Systems of Two First Order Linear ODEs Math 245 - Mathematics of Physics and Engineering I Lecture 9. Systems of Two First Order Linear ODEs January 30, 2012 Konstantin Zuev (USC) Math 245, Lecture 9 January 30, 2012 1 / 15 Agenda General Form

More information

Second In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 31 March 2011

Second In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 31 March 2011 Second In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 31 March 211 (1) [6] Give the interval of definition for the solution of the initial-value problem d 4 y dt 4 + 7 1 t 2 dy dt

More information

System Simulation using Matlab

System Simulation using Matlab EE4314 Fall 2008 System Simulation using Matlab The purpose of this laboratory work is to provide experience with the Matlab software for system simulation. The laboratory work contains a guide for solving

More information

OPTIMAL CONTROL. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 28

OPTIMAL CONTROL. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 28 OPTIMAL CONTROL Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 28 (Example from Optimal Control Theory, Kirk) Objective: To get from

More information

1.1 Differential Equation Models. Jiwen He

1.1 Differential Equation Models. Jiwen He 1.1 Math 3331 Differential Equations 1.1 Differential Equation Models Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math3331 Jiwen He, University of

More information

Practice Midterm Exam 1. Instructions. You have 60 minutes. No calculators allowed. Show all your work in order to receive full credit.

Practice Midterm Exam 1. Instructions. You have 60 minutes. No calculators allowed. Show all your work in order to receive full credit. MATH202X-F01/UX1 Spring 2015 Practice Midterm Exam 1 Name: Answer Key Instructions You have 60 minutes No calculators allowed Show all your work in order to receive full credit 1 Consider the points P

More information

AH Mechanics Checklist (Unit 1) AH Mechanics Checklist (Unit 1) Rectilinear Motion

AH Mechanics Checklist (Unit 1) AH Mechanics Checklist (Unit 1) Rectilinear Motion Rectilinear Motion No. kill Done 1 Know that rectilinear motion means motion in 1D (i.e. along a straight line) Know that a body is a physical object 3 Know that a particle is an idealised body that has

More information

Separable Differential Equations

Separable Differential Equations Separable Differential Equations MATH 6 Calculus I J. Robert Buchanan Department of Mathematics Fall 207 Background We have previously solved differential equations of the forms: y (t) = k y(t) (exponential

More information

Lecture 8: Ordinary Differential Equations

Lecture 8: Ordinary Differential Equations MIT-WHOI Joint Program Summer Math Review Summer 2015 Lecture 8: Ordinary Differential Equations Lecturer: Isabela Le Bras Date: 31 July 2015 Disclaimer: These notes are for the purposes of this review

More information

Physics 6010, Fall Relevant Sections in Text: Introduction

Physics 6010, Fall Relevant Sections in Text: Introduction Physics 6010, Fall 2016 Introduction. Configuration space. Equations of Motion. Velocity Phase Space. Relevant Sections in Text: 1.1 1.4 Introduction This course principally deals with the variational

More information

Announcements Wednesday, September 27

Announcements Wednesday, September 27 Announcements Wednesday, September 27 The midterm will be returned in recitation on Friday. You can pick it up from me in office hours before then. Keep tabs on your grades on Canvas. WeBWorK 1.7 is due

More information

Projectile Motion: Vectors

Projectile Motion: Vectors Projectile Motion: Vectors Ch. 5 in your text book Students will be able to: 1) Add smaller vectors going in the same direction to get one large vector for that direction 2) Draw a resultant vector for

More information

q = tan -1 (R y /R x )

q = tan -1 (R y /R x ) Vector Addition Using Vector Components = + R x = A x + B x B y R y = A y + B y R = (R x 2 + R y 2 ) 1/2 B x q = tan -1 (R y /R x ) Example 1.7: Vector has a magnitude of 50 cm and direction of 30º, and

More information

2-D Kinematics. In general, we have the following 8 equations (4 per dimension): Notes Page 1 of 7

2-D Kinematics. In general, we have the following 8 equations (4 per dimension): Notes Page 1 of 7 2-D Kinematics The problem we run into with 1-D kinematics, is that well it s one dimensional. We will now study kinematics in two dimensions. Obviously the real world happens in three dimensions, but

More information

Coordinate systems and vectors in three spatial dimensions

Coordinate systems and vectors in three spatial dimensions PHYS2796 Introduction to Modern Physics (Spring 2015) Notes on Mathematics Prerequisites Jim Napolitano, Department of Physics, Temple University January 7, 2015 This is a brief summary of material on

More information

Chapter 1: Introduction

Chapter 1: Introduction Chapter 1: Introduction Definition: A differential equation is an equation involving the derivative of a function. If the function depends on a single variable, then only ordinary derivatives appear and

More information

Dealing with Rotating Coordinate Systems Physics 321. (Eq.1)

Dealing with Rotating Coordinate Systems Physics 321. (Eq.1) Dealing with Rotating Coordinate Systems Physics 321 The treatment of rotating coordinate frames can be very confusing because there are two different sets of aes, and one set of aes is not constant in

More information

TRACKING AND DISTURBANCE REJECTION

TRACKING AND DISTURBANCE REJECTION TRACKING AND DISTURBANCE REJECTION Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 13 General objective: The output to track a reference

More information

Chap. 3 MATH Annalisa Quaini Office : PGH 662 Lecture : MWF 11AM-12PM Office hours : W 8AM-10AM

Chap. 3 MATH Annalisa Quaini Office : PGH 662 Lecture : MWF 11AM-12PM Office hours : W 8AM-10AM Chap. 3 MATH 1431-18319 Annalisa Quaini quaini@math.uh.edu Office : PGH 662 Lecture : MWF 11AM-12PM Office hours : W 8AM-10AM Daily quiz 6 is due on Friday at 11 AM. Exam 1 is coming - check the scheduler

More information

(A) Opening Problem Newton s Law of Cooling

(A) Opening Problem Newton s Law of Cooling Lesson 55 Numerical Solutions to Differential Equations Euler's Method IBHL - 1 (A) Opening Problem Newton s Law of Cooling! Newton s Law of Cooling states that the temperature of a body changes at a rate

More information

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS Basic Concepts Paul Dawkins Table of Contents Preface... Basic Concepts... 1 Introduction... 1 Definitions... Direction Fields... 8 Final Thoughts...19 007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx

More information

Motion Along a Straight Line (Motion in One-Dimension)

Motion Along a Straight Line (Motion in One-Dimension) Chapter 2 Motion Along a Straight Line (Motion in One-Dimension) Learn the concepts of displacement, velocity, and acceleration in one-dimension. Describe motions at constant acceleration. Be able to graph

More information

4.9 Anti-derivatives. Definition. An anti-derivative of a function f is a function F such that F (x) = f (x) for all x.

4.9 Anti-derivatives. Definition. An anti-derivative of a function f is a function F such that F (x) = f (x) for all x. 4.9 Anti-derivatives Anti-differentiation is exactly what it sounds like: the opposite of differentiation. That is, given a function f, can we find a function F whose derivative is f. Definition. An anti-derivative

More information

P - f = m a x. Now, if the box is already moving, for the frictional force, we use

P - f = m a x. Now, if the box is already moving, for the frictional force, we use Chapter 5 Class Notes This week, we return to forces, and consider forces pointing in different directions. Previously, in Chapter 3, the forces were parallel, but in this chapter the forces can be pointing

More information

20.6. Transfer Functions. Introduction. Prerequisites. Learning Outcomes

20.6. Transfer Functions. Introduction. Prerequisites. Learning Outcomes Transfer Functions 2.6 Introduction In this Section we introduce the concept of a transfer function and then use this to obtain a Laplace transform model of a linear engineering system. (A linear engineering

More information

Modeling and Experimentation: Compound Pendulum

Modeling and Experimentation: Compound Pendulum Modeling and Experimentation: Compound Pendulum Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin Fall 2014 Overview This lab focuses on developing a mathematical

More information

Review: control, feedback, etc. Today s topic: state-space models of systems; linearization

Review: control, feedback, etc. Today s topic: state-space models of systems; linearization Plan of the Lecture Review: control, feedback, etc Today s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest Once we have mastered

More information

Ordinary Differential Equations

Ordinary Differential Equations Ordinary Differential Equations Swaroop Nandan Bora swaroop@iitg.ernet.in Department of Mathematics Indian Institute of Technology Guwahati Guwahati-781039 Modelling a situation We study a model, a sort

More information

Ordinary Differential Equations (ODE)

Ordinary Differential Equations (ODE) Ordinary Differential Equations (ODE) Why study Differential Equations? Many physical phenomena are best formulated mathematically in terms of their rate of change. Motion of a swinging pendulum Bungee-jumper

More information

Things going in circles

Things going in circles Things going in circles Physics 211 Syracuse University, Physics 211 Spring 2019 Walter Freeman February 18, 2019 W. Freeman Things going in circles February 18, 2019 1 / 30 Announcements Homework 4 due

More information

Donald Fussell. October 28, Computer Science Department The University of Texas at Austin. Point Masses and Force Fields.

Donald Fussell. October 28, Computer Science Department The University of Texas at Austin. Point Masses and Force Fields. s Vector Moving s and Coputer Science Departent The University of Texas at Austin October 28, 2014 s Vector Moving s Siple classical dynaics - point asses oved by forces Point asses can odel particles

More information

Chapter 13 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Oscillatory Motion Pearson Education, Inc.

Chapter 13 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Oscillatory Motion Pearson Education, Inc. Chapter 13 Lecture Essential University Physics Richard Wolfson nd Edition Oscillatory Motion Slide 13-1 In this lecture you ll learn To describe the conditions under which oscillatory motion occurs To

More information

HOMEWORK 3 MA1132: ADVANCED CALCULUS, HILARY 2017

HOMEWORK 3 MA1132: ADVANCED CALCULUS, HILARY 2017 HOMEWORK MA112: ADVANCED CALCULUS, HILARY 2017 (1) A particle moves along a curve in R with position function given by r(t) = (e t, t 2 + 1, t). Find the velocity v(t), the acceleration a(t), the speed

More information

These notes are based mostly on [3]. They also rely on [2] and [1], though to a lesser extent.

These notes are based mostly on [3]. They also rely on [2] and [1], though to a lesser extent. Chapter 1 Introduction These notes are based mostly on [3]. They also rely on [2] and [1], though to a lesser extent. 1.1 Definitions and Terminology 1.1.1 Background and Definitions The words "differential

More information

MATH 333: Partial Differential Equations

MATH 333: Partial Differential Equations MATH 333: Partial Differential Equations Problem Set 9, Final version Due Date: Tues., Nov. 29, 2011 Relevant sources: Farlow s book: Lessons 9, 37 39 MacCluer s book: Chapter 3 44 Show that the Poisson

More information

Write these equations in your notes if they re not already there. You will want them for Exam 1 & the Final.

Write these equations in your notes if they re not already there. You will want them for Exam 1 & the Final. Tuesday January 24 Assignment 3: Due Friday, 11:59pm.like every Friday Pre-Class Assignment: 15min before class like every class Office Hours: Wed. 10-11am, 204 EAL Help Room: Wed. & Thurs. 6-9pm, here

More information

Math 322. Spring 2015 Review Problems for Midterm 2

Math 322. Spring 2015 Review Problems for Midterm 2 Linear Algebra: Topic: Linear Independence of vectors. Question. Math 3. Spring Review Problems for Midterm Explain why if A is not square, then either the row vectors or the column vectors of A are linearly

More information

This chapter covers all kinds of problems having to do with work in physics terms. Work

This chapter covers all kinds of problems having to do with work in physics terms. Work Chapter 7 Working the Physics Way In This Chapter Understanding work Working with net force Calculating kinetic energy Handling potential energy Relating kinetic energy to work This chapter covers all

More information

Chapter 9. Linear momentum and collisions. PHY 1124 Fundaments of Physics for Engineers. Michael Wong PHY1124 Winter uottawa.

Chapter 9. Linear momentum and collisions. PHY 1124 Fundaments of Physics for Engineers. Michael Wong PHY1124 Winter uottawa. Chapter 9 Linear momentum and collisions Michael Wong PHY1124 Winter 2019 PHY 1124 Fundaments of Physics for Engineers uottawa.ca https://uottawa.brightspace.com/d2l/home Goals 2 Chapter 9 Momentum and

More information

An Introduction to Differential Equations

An Introduction to Differential Equations An Introduction to Differential Equations Let's start with a definition of a differential equation. A differential equation is an equation that defines a relationship between a function and one or more

More information

Chapter 9 - Polarization

Chapter 9 - Polarization Chapter 9 - Polarization Gabriel Popescu University of Illinois at Urbana Champaign Beckman Institute Quantitative Light Imaging Laboratory http://light.ece.uiuc.edu Principles of Optical Imaging Electrical

More information

Physics 225 Relativity and Math Applications. Fall Unit 7 The 4-vectors of Dynamics

Physics 225 Relativity and Math Applications. Fall Unit 7 The 4-vectors of Dynamics Physics 225 Relativity and Math Applications Fall 2011 Unit 7 The 4-vectors of Dynamics N.C.R. Makins University of Illinois at Urbana-Champaign 2010 Physics 225 7.2 7.2 Physics 225 7.3 Unit 7: The 4-vectors

More information

Path integrals in quantum mechanics

Path integrals in quantum mechanics Path integrals in quantum mechanics Phys V3500/G8099 handout #1 References: there s a nice discussion of this material in the first chapter of L.S. Schulman, Techniques and applications of path integration.

More information

Ordinary Differential Equations I

Ordinary Differential Equations I Ordinary Differential Equations I CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Ordinary Differential Equations I 1 / 32 Theme of Last Few

More information

Chapter1. Ordinary Differential Equations

Chapter1. Ordinary Differential Equations Chapter1. Ordinary Differential Equations In the sciences and engineering, mathematical models are developed to aid in the understanding of physical phenomena. These models often yield an equation that

More information

Section 1.8/1.9. Linear Transformations

Section 1.8/1.9. Linear Transformations Section 1.8/1.9 Linear Transformations Motivation Let A be a matrix, and consider the matrix equation b = Ax. If we vary x, we can think of this as a function of x. Many functions in real life the linear

More information

Solutions to Homework 1, Introduction to Differential Equations, 3450: , Dr. Montero, Spring y(x) = ce 2x + e x

Solutions to Homework 1, Introduction to Differential Equations, 3450: , Dr. Montero, Spring y(x) = ce 2x + e x Solutions to Homewor 1, Introduction to Differential Equations, 3450:335-003, Dr. Montero, Spring 2009 problem 2. The problem says that the function yx = ce 2x + e x solves the ODE y + 2y = e x, and ass

More information

2. Two Dimensional Kinematics

2. Two Dimensional Kinematics . Two Dimensional Kinematics A) Overview We will begin by introducing the concept of vectors that will allow us to generalize what we learned last time in one dimension to two and three dimensions. In

More information

Solving Differential Equations: First Steps

Solving Differential Equations: First Steps 30 ORDINARY DIFFERENTIAL EQUATIONS 3 Solving Differential Equations Solving Differential Equations: First Steps Now we start answering the question which is the theme of this book given a differential

More information

Chapter 3 Acceleration

Chapter 3 Acceleration Chapter 3 Acceleration Slide 3-1 PackBack The first answer gives a good physical picture. The video was nice, and worth the second answer. https://www.youtube.com/w atch?v=m57cimnj7fc Slide 3-2 Slide 3-3

More information

Example force problems

Example force problems PH 105 / LeClair Fall 2015 Example force problems 1. An advertisement claims that a particular automobile can stop on a dime. What net force would actually be necessary to stop a 850 kg automobile traveling

More information

MATH 308 Differential Equations

MATH 308 Differential Equations MATH 308 Differential Equations Summer, 2014, SET 1 JoungDong Kim Set 1: Section 1.1, 1.2, 1.3, 2.1 Chapter 1. Introduction 1. Why do we study Differential Equation? Many of the principles, or laws, underlying

More information

POLE PLACEMENT. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 19

POLE PLACEMENT. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 19 POLE PLACEMENT Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 19 Outline 1 State Feedback 2 Observer 3 Observer Feedback 4 Reduced Order

More information

Physics H7A, Fall 2011 Homework 4 Solutions

Physics H7A, Fall 2011 Homework 4 Solutions Physics H7A, Fall 20 Homework 4 Solutions. (K&K Problem 2.) A mass m is connected to a vertical revolving axle by two strings of length l, each making an angle of 45 with the axle, as shown. Both the axle

More information

The Cross Product. In this section, we will learn about: Cross products of vectors and their applications.

The Cross Product. In this section, we will learn about: Cross products of vectors and their applications. The Cross Product In this section, we will learn about: Cross products of vectors and their applications. THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is a

More information

PHY321 Homework Set 2

PHY321 Homework Set 2 PHY321 Homework Set 2 1. [5 pts] Consider the forces from the previous homework set, F A ( r )and F B ( r ), acting on a particle. The force components depend on position r of the particle according to

More information

MA Ordinary Differential Equations

MA Ordinary Differential Equations MA 108 - Ordinary Differential Equations Santanu Dey Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 76 dey@math.iitb.ac.in March 26, 2014 Outline of the lecture Method

More information

First-Order Differential Equations

First-Order Differential Equations CHAPTER 1 First-Order Differential Equations 1. Diff Eqns and Math Models Know what it means for a function to be a solution to a differential equation. In order to figure out if y = y(x) is a solution

More information

Notes on numerical solution of differential equations

Notes on numerical solution of differential equations Notes on numerical solution of differential equations Some definitions, for those who don t know: A differential equation is any equation that relates a thing to its derivatives. For instance, Newton s

More information

Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories

Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 203 Outline

More information

First Order ODEs (cont). Modeling with First Order ODEs

First Order ODEs (cont). Modeling with First Order ODEs First Order ODEs (cont). Modeling with First Order ODEs September 11 15, 2017 Bernoulli s ODEs Yuliya Gorb Definition A first order ODE is called a Bernoulli s equation iff it is written in the form y

More information

Second Order Linear Equations

Second Order Linear Equations October 13, 2016 1 Second And Higher Order Linear Equations In first part of this chapter, we consider second order linear ordinary linear equations, i.e., a differential equation of the form L[y] = d

More information

Vectors and Coordinate Systems

Vectors and Coordinate Systems Vectors and Coordinate Systems In Newtonian mechanics, we want to understand how material bodies interact with each other and how this affects their motion through space. In order to be able to make quantitative

More information

Laplace Transforms Chapter 3

Laplace Transforms Chapter 3 Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important

More information

The Big Picture. Discuss Examples of unpredictability. Odds, Stanisław Lem, The New Yorker (1974) Chaos, Scientific American (1986)

The Big Picture. Discuss Examples of unpredictability. Odds, Stanisław Lem, The New Yorker (1974) Chaos, Scientific American (1986) The Big Picture Discuss Examples of unpredictability Odds, Stanisław Lem, The New Yorker (1974) Chaos, Scientific American (1986) Lecture 2: Natural Computation & Self-Organization, Physics 256A (Winter

More information

c 1 = y 0, c 2 = 1 2 y 1. Therefore the solution to the general initial-value problem is y(t) = y 0 cos(2t)+y sin(2t).

c 1 = y 0, c 2 = 1 2 y 1. Therefore the solution to the general initial-value problem is y(t) = y 0 cos(2t)+y sin(2t). Solutions to Second In-Class Exam Math 246, Professor David Levermore Tuesday, 29 October 2 ( [4] Give the interval of definition for the solution of the initial-value problem u t u + cos(5t 6+t u = et

More information

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 15 - Tues 20th Nov 2018 First and Higher Order Differential Equations

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 15 - Tues 20th Nov 2018 First and Higher Order Differential Equations ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 15 - Tues 20th Nov 2018 First and Higher Order Differential Equations Integrating Factor Here is a powerful technique which will work (only!)

More information

Physics 351 Monday, February 26, 2018

Physics 351 Monday, February 26, 2018 Physics 351 Monday, February 26, 2018 You just read the first half ( 10.1 10.7) of Chapter 10, which we ll probably start to discuss this Friday. The midterm exam (March 26) will cover (only!) chapters

More information

Old Dominion University Momentum and Collisions

Old Dominion University Momentum and Collisions m1 v1,b m2 v2,b = 0 before University Physics 226N/231N Old Dominion University Momentum and Collisions Dr. Todd Satogata (ODU/Jefferson Lab) satogata@jlab.org http://www.toddsatogata.net/2012-odu Friday,

More information

Bayesian Inference and the Symbolic Dynamics of Deterministic Chaos. Christopher C. Strelioff 1,2 Dr. James P. Crutchfield 2

Bayesian Inference and the Symbolic Dynamics of Deterministic Chaos. Christopher C. Strelioff 1,2 Dr. James P. Crutchfield 2 How Random Bayesian Inference and the Symbolic Dynamics of Deterministic Chaos Christopher C. Strelioff 1,2 Dr. James P. Crutchfield 2 1 Center for Complex Systems Research and Department of Physics University

More information