Archives

Differential Equations for Engineers: the Essentials Class 1 notes Agenda: Class 1 Administrative matters Course objectives Importance of DEs for engineers Lectures: Review of mathematical foundations Classes of differential equations Homework assignment 1 Administrative Matters Homework, tests, attendance and grading […]

Some Key Concepts & Terms Logarithm Exponential Derivative Integral Infinite series Imaginary numbers Matrix algebra Later in the course: Exponential and Logarithm Primitive definitions: ( = 10 multiplied times, an integer ) Relations: y x== 1010101010 x x log 10 […]

Differential Equations for Engineers: the Essentials Class 10 notes Agenda: Class 10 Reviewing Homework #6 Lectures: Second Order Linear nonhomogeneous ODEs: General solution Homework Assignment 10 Engineering example: automobile cruise control Another approach to solving linear time-invariant nonhomogeneous ODEs Engineering […]

Undetermined Coefficient Method: When It Doesn’t Work The Undetermined Coefficient Method doesn’t work when: The ODE is not linear time-invariant The input is not or some combination thereof The input is a fundamental solution of the ODE. ttett ktn  […]

Example: Cruise Control (8) Then Equation 2 becomes As we have just seen, the solution to Equation 3 can be written  sin2 2 2 2 gy dt dy dt yd −=++ Equation 3 −−= t tdgtKty 0 )sin)(()(  […]

Fundamental Solutions Theorem : If the functions are solutions to the Equation 1 with and their Wronskian .. .. 121 121 − − nn nn yyyy yyyy  )(),(),…,(),(121 tytytyty nn− 0)( =tg then any solution to Equation 1 with […]

Satellite Orbit Stability (3) We can convert the equations of motion into polar coordinates and obtain R  22 222 2 )( 2 2 )( 2 RR m RSC RR RR m RSC R R GM R D D  […]

Differential Equations for Engineers: the Essentials Class 11 notes Agenda: Class 11 Review Homework Assignment 9 Lectures: Second order linear time-invariant nonhomogeneous ODEs (1) Procedure and example for the kernel method Homework Assignment 11 (2) Example for trial and error […]

Homework Assignment 14 Review: Refresh your memory by skimming (the guide to the instructor’s or student’s choice of numerical integration computer programs.) Work : We will use the numerical integration computer program to explore aircraft flight performance. We will add […]

Homework Assignment 14: Example Engineering Application of ODEs Sounding Rocket Performance Phase 2 Homework Assignment 14 Read: Refresh your memory by skimming (the guide to the instructor’s or student’s choice of numerical integration computer programs.) Work : We will use […]

Differential Equations for Engineers: the Essentials Class 14 notes Agenda: Class 14 Working on a qualitative aspect of ODEs: system stability Lectures: Introduction to Laplace transforms Laplace transforms for solving initial value problems Computing Project Phase 2 Questions on Determining […]

Dirac Delta Function (5) Consider the ODE with zero initial conditions. The solution, by time domain methods, is )(… 2 2 2 1 1 1tya dt yd a dt yd a dt yd n n n n n n n […]

Response to (non-repeated) square wave input (10) Therefore     )2()( )()( 020 21 00 1 TtyusYeL TtyusYeL T sT T sT −= −= −− −− and )2()(2)()( 0200 TtyuTtyutyty TT −+−−= where, again, )sin)/(cos1( )( 1 )( […]

Differential Equations for Engineers: the Essentials Class 15 notes Agenda: Class 15 Answers to questions on system stability Lectures: Laplace transforms for solving nonhomogeneous ODEs Impulse and impulse response Homework Assignment 15 Response to a non-repeating square wave input Initial […]

Systems of First Order ODEs Linear Systems in State Space Format Linearization better defined Let be the known solution of a nonlinear nth order system defined by Let be the unknown solution of a slightly different nth order nonlinear ODE […]

Euler Method: Example (3) Second time step: Third time step 3.01.02.0 =+=+= ttt 172.9)1.0(224.5163.9)162.1(613.9)2/3exp( 421.5)1.0(224.5163.9)995.0(224.5)2cos( 11112 11112 =+−=+−= =−+=−+= tZVtVV tZVtZZ 2.01.01.0 12 =+=+= ttt 656.8)1.0(418.5172.9)350.1(172.9)2/3exp( 611.5)1.0(421.5172.9)980.0(421.5)2cos( 22223 22223 23 =+−=+−= =−+=−+= tZVtVV tZVtZZ Systems of First Order ODEs Review […]

Differential Equations for Engineers: the Essentials Class 16 notes Agenda: Class 16 Review of Computing Project #2 Lectures: Systems of First Order ODEs Nomenclature Homework Assignment 16 Numerical Methods Review of Matrix Algebra Linear Systems in State Space Format Systems […]

Differential Equations for Engineers: the Essentials Class 17 notes Agenda: Class 17 Review Homework Assignment 15 Lectures: Systems of First Order ODEs: Homework Assignment 17 Short Quiz #2 Linear Time-Invariant Example: Heat Transfer Systems of First Order ODEs Linear Time […]

Differential Equations for Engineers: the Essentials Class 18 notes Agenda: Class 18 Review of Short Quiz #2 Review of Homework Assignments 14 and 16 Lectures: Complex Eigenvalues and Eigenvectors Homework Assignment 21 (1) Electrical circuit (2) Aircraft longitudinal dynamics Objective […]

Vehicle Suspension System Model Definitions Position of vehicle centerline at right wheel relative to equilibrium position = 1 y = 2 y Position of vehicle centerline at left wheel relative to equilibrium position = cg y Position of vehicle center […]

Class 2 notes Differential Equations for Engineers: the Essentials Agenda: Class 2 First order linear differential equations: (1) Engineering example: RC circuit (2) General solution of homogeneous equation Homework Assignment 2 (3) In-class homogeneous problems (4) Example: Exposed water pipe […]

First Order Linear Differential Equations Example: Exposed water pipe in cyclical ambient temperature Air at temperature tTTT  sin 0 +=  Exposed Water Pipe in Cyclical Ambient Temperature Water at temperature TTTW+= d Wall thickness = Outer radius = […]

Differential Equations for Engineers: the Essentials Class 23 notes Agenda: Class 23 Review Homework Assignment 21 Lectures: Partial Differential Equations: Heat Equation: Boundary Value Problem Example Homework Assignment 23 Short Quiz #3 Partial Differential Equations Heat Equation: Boundary Value Example […]

Heat Equation: Boundary Value Example Consider a rod of length Lwhose ends are controlled to temperature 0 and to which is applied a point heat source at x = L/4, raising the temperature there to , and a point heat […]

Differential Equations for Engineers: the Essentials Class 24 notes Agenda: Class 24 Review Short Quiz #3 Review Homework Assignment 22 Lectures: Partial Differential Equations: Wave Equation Initial value problem example Boundary value problem example Partial Differential Equations Wave Equation Wave […]

Modified Bessel Functions The modified Bessel’s Equation of order mis Its solutions are the modified Bessel function of the first kind of order m 0)/1( 122 2 2 =+−+ yxm dx dy xdx yd which has finite and at the […]

Differential Equations for Engineers: the Essentials Class 25 notes Agenda: Class 25 Review Homework Assignment 23 Lectures: Partial Differential Equations: Power Series Solutions –Part 1 Bessel Functions Power Series Solutions Bessel function of first kind of order m Some general […]

Differential Equations for Engineers: the Essentials Class 26 notes Agenda: Class 26 Lectures: Partial Differential Equations: Potential Equation –Introduction Potential Equation –Boundary value problem example Beam Equation Partial Differential Equations Potential Equation: Introduction Potential Equation The potential equation (also called […]

Some Fundamental Questions About a Sounding Rocket How high does it go? What maximum speed does it achieve going up? What maximum speed does it achieve coming down? Deriving the Equations for Sounding Rocket Motion: Basic Approach Break the rocket […]

Differential Equations for Engineers: the Essentials Class 3 notes Agenda: Class 3 Reviewing Homework Assignment 1 Lectures: First order linear differential equations System viewpoint First order nonlinear differential equations Homework Assignment 3 General approach to solving separable nonlinear first order […]

Differential Equations for Engineers: the Essentials Class 4 notes Agenda: Class 4 Review homework assignment 2 Lectures: Separable first order ODEs: Example: Sounding rocket –phase 3 Possibility of Transforming Nonlinear to Linear Supplement: Differences Between Linear and Nonlinear ODEs in […]

Differential Equations for Engineers: the Essentials Class 5 notes Agenda: Class 5 Review homework assignment 3 Lecture: First order ODEs: Successive approximations (general case) Homework Assignment 5 In-class problems Existence and uniqueness of solutions First Order Differential Equations Successive approximations […]

Preliminaries: Lipschitz Condition (3) Note that, in the previous example, the Lipschitz constant K was given by the point in R where This is true more generally: max ),( =   y ytf If is continuous in R and […]

Example Engineering Application of ODEs A numerical integration project: Sounding Rocket Performance – Phase 1 Numerical Integration of ODEs Using the computer to calculate a solution time-step by time-step The engineering professional standard –how engineers deal in practice with most […]

Definition of Terms = = = = m V h  altitude (ft) speed (ft/sec) flight path angle (radians) mass (slugs) = = = = = L D a C C S g  acceleration of gravity (ft/sec^2) air density […]

LRC Circuit: Roots of Characteristic Equation LCL R L R r1 22 2 −              −= Have finished Case 1: two real roots Now consider Case 2: repeated […]

Differential Equations for Engineers: the Essentials Class 7 notes Agenda: Class 7 Review Homework Assignment 5 Lecture: Second order linear time-invariant homogeneous ODEs: LRC circuit Characteristic equation In-class example problems Homework Assignment 7 Second Order Linear Time-Invariant Homogeneous ODEs Example: […]

Differential Equations for Engineers: the Essentials Class 9 notes Agenda: Class 9 Review of Test #1 Review Homework Assignment 7 Lecture: Second order linear time-invariant homogeneous ODEs: In-class problems Homework Assignment 9 LRC Circuit: complex roots Fundamental solutions Second Order […]

LRC Circuit: Complex Roots Summary(3) The results of this example extend to any ODE of the form 02 2 0 2 2 =++ y dt dy dt yd  when The solution is where   0 )sin()cos()( 21 tectecty […]

Differential Equations for Engineers: the Essentials Supplement to Class 25 notes Partial Differential Equations Power Series Solutions: Bessel Function of Second Kind of Order Zero Bessel Functions: Second Solution The second fundamental solution to the Bessel equation of order m, […]

Differential Equations for Engineers: the Essentials Supplement to Class 4 Notes Contents of Supplement Differences Between Linear and Nonlinear ODEs in Their Input / Output Response Example: Nonlinear Circuit Differences Between Linear and Nonlinear ODEs in Their Input / Output […]

Differential Equations for Engineers: the Essentials Supplement to Class 14B Homework Assignment 14: Example Engineering Application of ODEs Aircraft Flight Performance, Phase 2 Homework Assignment 14 Review: Refresh your memory by skimming (the guide to the instructor’s or student’s choice […]