DIFFERENTIAL EQUATIONS MTH401 Virtual University of Pakistan Knowledge beyond the boundaries Table of Contents 1 Introduction ................................................................................................................. 1 2 Fundamentals .............................................................................................................. 3 2.1 Elements of the Theory ........................................................................................ 3 2.2 Specific Examples of ODE’s ............................................................................... 3 2.3 The order of an equation ...................................................................................... 4 2.4 Ordinary Differential Equation ............................................................................ 4 2.5 Partial Differential Equation ................................................................................ 4 2.6 Results from ODE data ........................................................................................ 4 2.7 BVP Examples ..................................................................................................... 5 2.8 Properties of ODE’s ............................................................................................. 5 2.9 Superposition ........................................................................................................ 5 2.10 Explicit Solution ............................................................................................... 5 2.11 Implicit Solution ............................................................................................... 5 3 Separable Equations .................................................................................................... 6 3.1 Solution steps of Separable Equations ................................................................. 6 3.2 Exercise .............................................................................................................. 13 4 Homogeneous Differential Equations ....................................................................... 15 4.1 Method of Solution............................................................................................. 16 4.2 Equations reducible to homogenous form ......................................................... 17 4.2.1 Case 1 ......................................................................................................... 18 4.2.2 Case 2 ......................................................................................................... 18 4.3 Exercise .............................................................................................................. 20 5 Exact Differential Equations .................................................................................... 22 5.1 Method of Solution ............................................................................................ 22 5.2 Exercise .............................................................................................................. 26 6 Integrating Factor Technique .................................................................................... 28 6.1 Case 1 ................................................................................................................ 28 6.2 Case 2 ................................................................................................................ 29 6.3 Case 3 ................................................................................................................ 29 6.4 Case 4 ................................................................................................................. 29 6.5 Exercise .............................................................................................................. 36 7 First Order Linear Equations .................................................................................... 37 7.1 Method of solution ............................................................................................. 37 7.2 Exercise .............................................................................................................. 41 8 Bernoulli Equations .................................................................................................. 43 8.1 Method of solution ............................................................................................. 43 8.2 Exercise .............................................................................................................. 47 8.3 Substitutions ....................................................................................................... 48 8.4 Exercise .............................................................................................................. 52 9 Solved Problems ....................................................................................................... 53 10 Applications of First Order Differential Equations ............................................... 69 10.1 Orthogonal Trajectories ................................................................................. 69 10.2 Orthogonal curves .......................................................................................... 71 10.3 Orthogonal Trajectories (OT) ........................................................................ 71 10.3.1 Method of finding Orthogonal Trajectory .............................................. 72 10.4 Population Dynamics ..................................................................................... 77 11 Radioactive Decay ................................................................................................. 80 11.1 Newton's Law of Cooling ............................................................................... 82 11.2 Carbon Dating ................................................................................................ 84 12 Applications of Non-linear Equations ................................................................... 86 12.1 Logistic equation ............................................................................................ 86 12.1.1 Solution of the Logistic equation ................................................................ 86 12.1.2 Special Cases of Logistic Equation............................................................. 87 12.1.3 A Modification of LE.................................................................................. 88 12.2 Chemical reactions ......................................................................................... 89 12.3 Miscellaneous Applications ............................................................................ 92 13 Higher Order Linear Differential Equations ............................................................. 94 13.1 Preliminary theory .......................................................................................... 94 13.2 Initial -Value Problem .................................................................................... 94 13.2.1 Solution of IVP ........................................................................................... 95 13.3 Theorem ( Existence and Uniqueness of Solutions) ....................................... 95 13.4 Boundary-value problem (BVP) ..................................................................... 97 13.4.1 Solution of BVP .......................................................................................... 97 13.4.2 Possible Boundary Conditions .................................................................... 98 13.5 Linear Dependence ....................................................................................... 100 13.6 Linear Independence ..................................................................................... 100 13.6.1 Case of two functions................................................................................ 100 13.7 Wronskian ..................................................................................................... 102 13.8 Theorem (Criterion for Linearly Independent Functions) ............................ 102 13.9 Exercise ........................................................................................................ 104 14 Solutions of Higher Order Linear Equations .......................................................... 106 14.1 Preliminary Theory ....................................................................................... 106 14.2 Superposition Principle................................................................................. 106 14.3 Linear Independence of Solutions ................................................................ 109 14.4 Fundamental Set of Solutions ....................................................................... 109 14.4.1 Existence of a Fundamental Set ................................................................ 110 14.5 General Solution-Homogeneous Equations.................................................. 110 14.6 Non-Homogeneous Equations ...................................................................... 112 14.7 Complementary Function ............................................................................. 112 14.8 General Solution of Non-Homogeneous Equations ..................................... 113 14.9 Superposition Principle for Non-homogeneous Equations .......................... 114 14.10 Exercise .......................................................................................................