2.1
                         Math 3331 Differential Equations
                              2.1 Differential Equations and Solutions
                                                          Jiwen He
                                  Department of Mathematics, University of Houston
                                                  jiwenhe@math.uh.edu
                                         math.uh.edu/∼jiwenhe/math3331
        Jiwen He, University of Houston                 Math 3331 Differential Equations                  Summer, 2014       1 / 20
                                                            2.1       Definition of ODE Solutions IVP Geometric Interp. Exercises
                                          2.1 ODE and Solutions
                   Definition of First Order ODE
                           Normal Form of ODE
                   Solutions of ODE
                           General Solution and Solution Curves
                           Particular Solution
                   Initial Value Problem
                           Solution of IVP
                           Interval of Existence
                   Geometric Interpretation of ODE
                           Direction Field
                           Geometric interpretation of Solutions
                           Numerical Solution of IVP
                   Worked out Examples from Exercises:
                           2.4, 2.13, 2.19
        Jiwen He, University of Houston                 Math 3331 Differential Equations                  Summer, 2014       2 / 20
                                                            2.1       Definition of ODE Solutions IVP Geometric Interp. Exercises
     Formal Definition of ODE
           Definition of ODE
           ODEis an equation involving an unknown function y of a single
           variable t together with one or more of its derivatives y′, y′′, etc.
           First Order ODE: General (Implicit) Form
           First order ODEs often arise naturally in the form
                                                      φ(t,y,y′) = 0,
           Example
                                                       t +4yy′ = 0.
           This form is too general to deal with, and we will find it necessary
           to solve equation for y′ to place it into “normal form”
                                                          y′ = − t
                                                                      4y
        Jiwen He, University of Houston                 Math 3331 Differential Equations                  Summer, 2014       3 / 20
                                                            2.1       Definition of ODE Solutions IVP Geometric Interp. Exercises
     Normal Form of ODE
           Normal Form
           Afirst-order ODE of the form
                                                         y′ = f (t,y)
           is said to be in normal form.
           Examples
                                                        y′ = y −t
                                                        y′ = −2ty
                                                        y′ = y2
                                                        y′ = cos(t y)
        Jiwen He, University of Houston                 Math 3331 Differential Equations                  Summer, 2014       4 / 20