Ordinary Differential Equations: A Systems Approach

1 Ordinary Differential Equations: A Systems ApproachBruce P. ConradNovember 24, 20101c 2010 Bruce P. Conrad2 Chapter 1 First-Order Equations34 CHAPTER 1. FIRST-ORDER IntroductionA Differential equation is a relation involving an unknown function andsome of its derivatives. For example,dydt=y+etis a Differential equation that asks for a function,y=f(t), whose derivativeis equal to the function pluset. By differentiating, you can verify that afunction such asy=tetmeets this equations are a source of fascinating mathematical prob-lems, and they have numerous mathematical model is a mathematical construction, such as a differ-ential equation, that simulates a natural or engineering phenomenon.

2 Mostapplications of Differential equations take the form of mathematical mod-els. For example, consider the problem of determining the velocityvof afalling object.~?vNewton s second law of motion tells us that the net force on the object isequal to the product of its mass,m, and its acceleration,dvdt. This law is adifferential equation,mdvdt=F,Ignoring air resistance, for an object falling close to the earth s surface theforce isF=mg, directed downward, wheregis approximately metersper second per second. Thus the Differential equationmdvdt=mgis a mathematical model corresponding to a falling solve the Differential equation, cancel the mass and note thatvis anantiderivative of the constantg; thusv=gt+C, whereCis an INTRODUCTION5We have just solved a Differential equation: The solution is not a singlefunction, but a family of functions depending on an arbitrary determine the equation of motion of this particular falling object, weneed to refine our example, an initial condition, specifying the velocity whent=0,will determine the equation of motion.

3 If we assume that the object wasfalling from rest, so thatv=0 whent=0, then we know that the solutionthat we seek has the propertyv(0) =0. We have knowledge of the starting,or initial value ofv. It is easy to infer from this initial condition that theconstantCis equal to 0, andv= if wedon tignore air resistance? It is known that the force of air re-sistance is directed oppositely to the motion, with magnitude proportionalto the square of the velocity (assuming the velocity is much less than thespeed of sound). In other words, air resistance=k v2, where the constantof proportionalitykis the drag coefficient). Combining air resistance andgravitation, we obtain the Differential equation modeldvdt=g kmv2.

4 ( )It may be tempting to integrate as we did before:v= (g kmv2) the unknown function vappears in the integrand here, there is noway to calculate this integral without first knowing the answer! Our objec-tive is to manage this order of a Differential equation is the order of the high-est derivative of the unknown function occurring in the equation. The dif-ferential equations describing the velocity of a falling object that we justconsidered above were first order. In the related second order equation,y =g, the unknown function represented by the variableyis the distancethe object has fallen. The velocity would bev=y . Including air resistance,we gety =g k(y )2/m, another second order Differential equation involving only derivatives with respect to a sin-gle independent variable is called anordinary Differential equation, orODE.

5 The falling body models that we just considered are ODEs, in whichthe independent variable ist. A Differential equation that involves partialderivatives with respect to two or more independent variables is a partial6 CHAPTER 1. FIRST-ORDER equations Differential equation, or PDE. As an example, here is the second order PDEthat models the vibration of a guitar string: 2y t2 c2 2y x2= unknown functionyrepresents the displacement of a point on thestringxcentimeters from the bridge at timet, andcis a constant relatedto the tension and density of the any Differential equation, a dependent variable is a variable that rep-resents an unknown function. A solution of a Differential equation is afunction that can be substituted for the dependent variable to produce ODEdydt=ky( )can be as a model for population growth, compound interest on savings accounts,etc.

6 (see section ). Show that y=Cekt,where C is an arbitrary constant, is asolution of this well. Henceequation ( ) becomes an equations typically have infinite families of solutions, butwe often need just one solution from the family. We refer to a single solutionof a Differential equation as a particular solution to emphasize that it is oneof a general solution of a Differential equation is the family of all itssolutions. The general solution of an ODE on an interval(a,b)is a familyof all solutions that are defined at every point of the interval(a,b). Findingthe general solution of an ODE requires two steps:calculationandverifica-tion. The calculation step is exemplified by our solution of the falling bodyequationv =g.

7 Unless a mistake was made in the integration, the familyof solutions we found will satisfy the verification step is to show that all solutions of the ODE belong tothis family. The following theorem from calculus is useful for this purpose:Theorem (Equal derivatives theorem)Let f1(t)and f2(t)be defined anddifferentiable on an interval(a,b)(infinite endpoints are permitted), and INTRODUCTION7that f 1(t) =f 2(t)for all t (a,b).Then there is a constant C such that f1(t) =f2(t) +C for all t (a,b).Example that the family of solutions v=gt+C for y =g is thegeneral (t) = (t)is another solution of the differentialequation, theny 1(t) =y 0(t) = follows from the equal derivativestheorem thaty1(t) y0(t) + other words,every solution of the ODEbelongs to the familyy=gt+ ODEs typically have families of solutions, they are frequentlycoupled with additional information (called constraints) to single out a so-lution of interest.

8 The constraint that we have already encountered, andshall frequently encounter in the future is an initial condition, specifyingthe value of the solution at an initial time. An ODE coupled with an initialcondition is called an initial value problem, or IVP. The motion of a bodyfalling from rest with air resistance would be modeled by the IVP,dvdt=g kv2;v(0) = by computerThere are many techniques available to find so-lutions of ODEs. However, many ODEs have solutions that can t be ex-pressed in terms of the familiar elementary functions you worked with incalculus courses. Since the invention of the first computer, solutions of in-tractable ODEs and PDEs have been calculated by calculations are done bynumericalmeans; that is, the computerdoes not work with a formula for a solution, but calculates a table of valuesthat give a close approximation of a solution of the Differential equationfrom the Differential equation itself.

9 As a simple example, consider theODE of the formy =f(t). The solution,y= f(t)dt+C,is helpful if a formula for the antiderivative off(t)is available. Withoutsuch a formula, we might turn to a numerical method of evaluating theintegral, such as the rectangle rule, the trapezoidal rule, or Simpson s methods for solving ODEs of the formy =f(t,y), where theright side involvesyas well ast, are generalizations of these rules. 8 CHAPTER 1. FIRST-ORDER EQUATIONSA numerical method cannot compute a family of solutions; it can onlyapproximate one solution at a time. The user is expected to provide con-straints to single out the solution to be approximated. A program that isdesigned to approximate solutions of ODEs with initial conditions as theconstraints is called an IVP there are programs that are primarily IVP solvers, computer al-gebra Systems (CAS), such asMaple, Mathematica, andMatlabinclude sub-routines that are IVP solvers, as well as the capability to find a formula forthe general solution of practically any Differential equation for which thereis an established method of solution.

10 These routines follow rules for ma-nipulating formulas instead of performing numerical calculations. Thereare advanced calculators that incorporate IVP solvers and CAS as study of an ODEF igure displays the graphs of severalsolutions of the ODEv =g kv2representing the motion of a falling objectwith air resistance. These graphs appear to have a common asymptote,v=100. We can explain this feature as follows: Letv = g/k, so thatg=kv2 . The ODE can be rewritten asv =k(v2 v2)Forv<v we see thatv >0. This means thatvis increasing. Whenv>v ,v <0, andvis decreasing. In both cases,vtends towardv , as infigure explanation of the behavior of solutions of a given ODE obtained byanalyzing the equation itself, without referring to a formula for the generalsolution, is called a qualitative study.