REVIEW: First Order Differential 
                   Equations
                                                Ahmed Kaffel
                                   Dept of Mathematical & Statistical Sciences
                                              Marquette University
                                              Miwaukee, Wisconsin
                                       Email: ahmed.kaffel@marquette.edu
                                      https://www.mscsnet.mu.edu/ ahmed/
            Ahmed Kaffel (ahmed.kaffel@marquette.edu)      Marquette University                         1/38
                                                April 6, 2020
     Outline
      1 Introduction
      2 Classification of Differential Equations by:
                Type
                Order
                Linearity
      3 Applications of Differential Equations
      4 Solving first order linear Differential Equations using integrating factor.
      5 Solving Separable Differential Equations.
      6 Solving Bernoulli’s equations.
      7 Conclusions
            Ahmed Kaffel (ahmed.kaffel@marquette.edu)      Marquette University                         2/38
   Introduction
     What is a Differential Equation?
     Introduction
          What is a Differential Equation?
        Introduction
          Definition (Differential Equation)
               Differential equations frequently arise in modeling
          An equation that contains derivatives of one or more unknown
               situations
          functions with respect to one or more independent variables is
               They describe population growth, chemical reactions, heat
          said to be a differential equation.
               exchange, motion, and many other applications
               Differential equations are continuous analogs of discrete
                 The classical example is Newton’s Law of motion
               dynamical systems
                       The mass of an object times its acceleration is equal to the
                       sum of the forces acting on that object
                       Acceleration is the first derivative of velocity or the second
                       derivative of position
                 In biology, a differential equation describes a growth rate, a
                 reaction rate, or the change in some physiological state
            Ahmed Kaffel (ahmed.kaffel@marquette.edu)      Marquette University                         3/38
     What is a Differential Equation?
     What is a Differential Equation?
         What is a Differential Equation?
         Definition (Differential Equation)
         An equation that contains derivatives of one or more unknown
         functions with respect to one or more independent variables is
         said to be a differential equation.
                Example :  y ' = 4 y+ 2t -1
            Ahmed Kaffel (ahmed.kaffel@marquette.edu)      Marquette University                         4/38