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MTHE / MATH 237 Differential Equations for Engineering Science i Queen’s University Mathematics and Engineering and Mathematics and Statistics MTHE / MATH 237 Differential Equations for Engineering Science Supplemental Course Notes Serdar Y¨uksel November 26, 2015 ii This document is a collection of supplemental lecture notes used for MTHE / MATH 237: Differential Equations for Engineering Science. Serdar Y¨uksel Contents 1 Introduction to Differential Equations ................................................... .......... 1 1.1 Introduction.................................... ............................................ 1 1.2 Classification of DifferentialEquations ............ ............................................. 1 1.2.1 OrdinaryDifferentialEquations ................. ........................................ 2 1.2.2 PartialDifferentialEquations .................. ......................................... 2 1.2.3 HomogeneousDifferentialEquations .............. ...................................... 2 1.2.4 N-thorderDifferentialEquations................ ........................................ 2 1.2.5 LinearDifferentialEquations ................... ........................................ 2 1.3 SolutionsofDifferentialequations ................ ............................................. 3 1.4 DirectionFields................................. ............................................ 3 1.5 Fundamental Questions on First-Order Differential Equations ...................................... 4 2 First-Order Ordinary Differential Equations ................................................... .... 5 2.1 Introduction.................................... ............................................ 5 2.2 ExactDifferentialEquations ...................... ............................................ 5 2.3 MethodofIntegratingFactors ...................... ........................................... 6 2.4 SeparableDifferentialEquations.................. ............................................. 7 2.5 Differential Equations with Homogenous Coefficients . ........................................... 7 2.6 First-Order Linear Differential Equations. .............................................. 8 2.7 Applications.................................... ............................................ 8 3 Higher-Order Ordinary Linear Differential Equations .............................................. 9 3.1 Introduction.................................... ............................................ 9 3.2 Higher-OrderDifferentialEquations............... ............................................. 9 3.2.1 LinearIndependence ............................ ...................................... 10 3.2.2 WronskianofasetofSolutions .................... ..................................... 10 iv Contents 3.2.3 Non-HomogeneousProblem........................ .................................... 11 4 Higher-Order Ordinary Linear Differential Equations .............................................. 13 4.1 Introduction.................................... ............................................ 13 4.2 Homogeneous Linear Equations with Constant Coefficients ........................................ 13 4.2.1 L(m) withdistinctRealroots.............................. ............................. 13 4.2.2 L(m) withrepeatedRealroots.............................. ............................ 14 4.2.3 L(m) withComplexroots................................... ........................... 14 4.3 Non-Homogeneous Equations and the Principle of Superposition ................................... 14 4.3.1 MethodofUndeterminedCoefficients............... ..................................... 15 4.3.2 VariationofParameters ......................... ....................................... 16 4.3.3 ReductionofOrder.............................. ...................................... 17 4.4 Applicationsof Second-OrderEquations ............. ........................................... 18 4.4.1 Resonance..................................... ...................................... 19 5 Systems of First-Order Linear Differential Equations ............................................... 21 5.1 Introduction.................................... ............................................ 21 5.1.1 Higher-Order Linear Differential Equations can be reducedtoFirst-OrderSystems .............. 22 5.2 TheoryofFirst-OrderSystems ...................... .......................................... 23 6 Systems of First-Order Constant Coefficient Differential Equations ................................... 25 6.1 Introduction.................................... ............................................ 25 6.2 HomogeneousCase................................. ......................................... 25 6.3 HowtoComputetheMatrixExponential................ ........................................ 26 6.4 Non-HomogeneousCase ............................. ........................................ 27 6.5 RemarksontheTime-VaryingCase .................... ........................................ 28 7 Stability and Lyapunov’s Method ................................................... .............. 29 7.1 Introduction.................................... ............................................ 29 7.2 Stability ....................................... ............................................ 29 7.2.1 LinearSystems................................. ...................................... 29 7.2.2 Non-LinearSystems............................. ...................................... 30 7.3 Lyapunov’sMethod................................ .......................................... 30 8 Laplace Transform Method ................................................... ................... 33 8.1 Introduction.................................... ............................................ 33 Contents v 8.2 Transformations................................. ............................................ 33 8.2.1 LaplaceTransform.............................. ...................................... 33 8.2.2 PropertiesoftheLaplaceTransform ............... ...................................... 34 8.2.3 Laplace Transform Method for solving Initial Value Problems................................ 34 8.2.4 InverseLaplaceTransform ....................... ...................................... 35 8.2.5 Convolution:.................................. ....................................... 35 8.2.6 StepFunction .................................. ...................................... 35 8.2.7 Impulseresponse ............................... ...................................... 35 9 On Partial Differential Equations and Separation of Variables ....................................... 37 10 Series Method ................................................... ............................... 39 10.1 Introduction................................... ............................................. 39 10.2 TaylorSeriesExpansions ......................... ............................................ 39 11 Numerical Methods ................................................... .......................... 41 11.1 Introduction................................... ............................................. 41 11.2 NumericalMethods............................... ........................................... 41 11.2.1 Euler’sMethod................................ ....................................... 41 11.2.2 ImprovedEuler’sMethod ........................ ...................................... 42 11.2.3 Including the Second Order Term in Taylor’s Expansion..................................... 42 11.2.4 OtherMethods ................................. ...................................... 42 A Brief Review of Complex Numbers ................................................... ............. 43 A.1 Introduction.................................... ............................................ 43 A.2 Euler’s Formula and Exponential Representation . ........................................... 43 B Similarity Transformation and the Jordan Canonical Form .......................................... 47 B.1 SimilarityTransformation ........................ ............................................ 47 B.2 Diagonalization................................. ............................................ 47 B.3 JordanForm...................................... .......................................... 47 1 Introduction to Differential Equations 1.1 Introduction Many phenomena in engineering, physics and broad areas of applied mathematics involve entities which change as a function of one or more variables. The movement of a car along a road, the propagation of sound and waves, the path an airplane takes, the amount of charge in a capacitor in an electrical circuit, the way a cake taken out from a hot oven and left in room temperature cools down and many such processes can all be modeled and described as differential equations. In class we discussed a number of examples such as the swinging pendulum, a falling object with drag and an electrical circuit involving resistors and capacitors. Let us make a definition. Definition 1.1.1 A differential equation is an equation that relates an unknown function and one or more of its derivatives of with respect to one or more independent variables. For instance, the equation dy = 5x dx − relates the first derivative of y with respect to x, with x. Here x is the independent variable and y is the unknown function (dependent variable). In this course, we will learn how to solve and analyze
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