Lecture 01 Ordinary Differential Equations Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 10 Something to Think About… How are the laws of nature formulated mathematically? Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 11 Differential Equations Physical laws of nature involve derivates (rates). As such, they are formulated as differential equations. Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 12 Definition A differential equation is an equation for a missing function (or collection of functions) in terms of the derivatives of those functions. Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 13 The Modeling Process The DE is part of the modelling process: • Formulate a physical mathematical terms • The result is often a DE • Solve the Equation • Interpret the results. Fall 2021 problem Applied Mathematics Lecture 01: Ordinary Differential Equations in 14 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 15 Solving the differential equation means finding an explicit function such that when it substituted back into the DE, the equation becomes an identity. Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 16 Unlimited Population Growth The rate at which a population of a species grows at a certain time is proportional to the total population at that time. Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 17 Newton’s Law of Cooling The rate at which a bodies temperature changes is proportional to the difference between the body temperature and the temperature of the surrounding medium Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 18 Draining of a Tank Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 19 A Vibrating Mass Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 20 It is important to remember that the differential equation is an expression of a physical situation (at least, in engineering). Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 21 Some Calculus Reminders Consider the function: The derivative is: Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 22 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 23 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 24 The Chain Rule Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 25 Types of DEs • Ordinary differential equation (ODE): Derivatives are with respect to a single independent variable • Partial differential equation (PDE): Derivatives are with respect to two or more independent variables Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 26 Orders of DEs The order of an ODE or PDE is the order of the highest derivative in the equation Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 27 Forms Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 28 Linearity Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 29 Characteristics of Liner Eqns Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 30 Non-Linear Equations Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 31 Solution of an ODE Any function , defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into an nth-order ODE reduces the equation to an identity • Interval I can be an open interval (a, b), a closed interval [a, b], an infinite interval (a, ), etc. • A solution of a differential equation that is identically zero on an interval I is a trivial solution Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 32 Solution Verification Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 33 The Solution Curve The graph of a solution of an ODE is a solution curve and it is continuous on its interval I while the domain of may differ from the interval I Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 34 An explicit solution is one in which the dependent variable is expressed solely in terms of the independent variable and constants • G(x,y) = 0 is an implicit solution if at least one function exists that satisfies the relation G and the ODE on I. Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 35 Families of Solutions Similar to integration, we usually obtain a solution to a first-order differential equation containing an arbitrary constant c A solution with a constant c represents a set of solutions, called a one-parameter family of solutions An n-parameter family of solutions solves an nth-order differential equation Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 36 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 37 Initial Value Problems Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 38 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 39 Example 1.1 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 40 Existence & Uniqueness When solving an IVP, consider 2 fundamental questions: • Does a solution exist? • If a solution exists, is it unique? Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 41 Existence: Does the differential equation possess solutions and do any of the solution curves pass through the point (x0, y0)? Uniqueness: When can we be certain there is precisely one solution curve passing through the point (x0, y0)? Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 42 From Ben-Dor, Shock Wave Reflection Phenomena Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 43 Example 1.2 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 44 A Theorem Let R be a rectangular region in the x-y plane that contains the point (x0,y0) in it’s interior. If And Are continuous on R, then there exists some interval I and a unique function y(x) defined on I that is a solution to the initial value problem. Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 45 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 46 Visualizing Solution Curves Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 47 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 48 Consider Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 49 Example 1.3 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 50 Useful Websites https://aeb019.hosted.uark.edu/dfield.html https://www.monroecc.edu/faculty/paulseeburger/ca lcnsf/DirectionField/ https://www.desmos.com/calculator/jabrxtyje0 https://homepages.bluffton.edu/~nesterd/apps/slopef ields.html Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 51 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 52 Unconstrained Growth Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 53 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 54 Autonomous Fall 2021 st 1 Order DEs Applied Mathematics Lecture 01: Ordinary Differential Equations 55 Limited Growth Model Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 56 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 57 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 58 Consider… Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 59 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 60 Solution Curves Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 61 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 62 Example 1.4 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 63 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 64 Sources and Sinks Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 65 Separable Equations Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 66 Example 1.5 - IVP Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 67 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 68 Example 1.6 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 69 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 70 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 71 Linear Equations Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 72 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 73 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 74 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 75 Fall 2021 Applied Mathematics Lecture 01: Ordinary Differential Equations 76