Differential Equations Brannan Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved. Chapter 01: Introduction

Chapter 1 Introduction In this introductory chapter we try to give perspective to your study of differential equations. We first formulate several problems to introduce terminology and to illustrate some of the basic ideas that we will return to frequently in the remainder of the book. We introduce three major methods—geometrical, analytical, and numerical—for investigating the solutions of these problems. We then discuss several ways of classifying differential equations in order to provide organizational structure for the book.

Chapter 1 Introduction 1.1 Mathematical Models, Solutions, and Direction Fields 1.2 Linear Equations: Method of Integrating Factors 1.3 Numerical Approximations: Euler’s Method 1.4 Classification of Differential Equations

1.1 Mathematical Models, Solutions, and Direction Fields Differential Equations – Real world relations involving rates at which one variable, say y, may change with respect to another variable, t. Most often, these relations take the form of equations containing y and certain of the derivatives y', y'' , . . . , y(n) of y with respect to t. The resulting equations are then referred to as differential equations.

Mathematical Models A differential equation that describes some physical process is referred to as a mathematical model of the process. Ex: Heat Transfer: Newton’s Law of Cooling. du/dt= −k(u − T ), or u' = −k(u − T ).

Terminology u' = −k(u − T ) - time t is an independent variable, - temperature u is a dependent variable because it depends on t, - k and T0 are parameters in the model. The equation is an ordinary differential equation because it has one, and only one, independent variable. A solution to above equation is a differentiable function u = φ(t) that satisfies the equation.

Terminology (Ctd.) All possible solutions to diff. eq. is called the general solution to the equation. The geometrical representation of the general solution is an infinite family of curves in the tu- plane called integral curves. An initial condition to the differential equation is a value of u for a particular t, such as u(0)=70. Diff eq. together with the initial condition forms an initial value problem.

Several integral curves of u' = −1.5(u − 60). The general solution to This diff. eq. is u = 60 + ce−1.5t . c=10 is the solution satisfying initial condition u(0)=70.

The Direction Field for y' = f(t,y) A solution of dy/dt = f (t, y) is a differentiable function y = φ(t) such that φ'(t) = f (t, φ(t)). Hence f (t, y) specifies the slope of a solution passing through the point y at time t. This information can be displayed geometrically by drawing a line segment with slope f (t, y) at the point (t, y). A direction field (or Slope filed) consists of a large number of such line segments, usually drawn on a rectangular grid. A direction field provides a geometric representation of the flow of solutions of the differential equation.

Direction field and equilibrium solution u = 60 for u' = −1.5(u − 60). A solution that does not change with time is called the equilibrium solution.

Example - Heating and Cooling of a Building Equation: du/dt + ku = kT0 + kA sin ωt Solution: u = T0 + kA/(k2 + ω2) ·(k sin ωt − ω cos ωt) + ce−kt Example: Let k = 1.5 day−1, T0 = 60◦F, A = 15◦F, and ω = 2π be given. Then, du/dt + 1.5u = 90 + 22.5 sin 2πt (25) with corresponding general solution u = 60 + 22.5/(2.25 + 4π2)· (1.5 sin 2πt − 2π cos 2πt) + ce−1.5t .

Example - Heating and Cooling of a Building (Ctd) The steady-state solution is the solution to the equation As t  ∞. In this example, it is: U(t) = 60 + (22.5/2.25 + 4π2)· (1.5 sin 2πt − 2π cos 2πt)

1.2 Linear Equations: Method of Integrating Factors DEFINITIONS: A differential equation that can be written in the form (standard form) dy/dt+ p(t)y = g(t) is said to be a first order linear equation in the dependent variable y. If h(t)=0, it is said to be homogeneous; otherwise, the equation is nonhomogenous. If powers of y are there equation becomes nonlinear.

The Method of Integrating Factors for Solving y' + p(t)y = g(t). 1. Put the equation in standard form: y' + p(t)y = g(t). 2. Calculate the integrating factor μ(t) = e∫p(t)dt 3. Multiply the equation by μ(t) and write it in the form [μ(t)y]' = μ(t)g(t). 4. Integrate this equation to obtain μ(t)y = ∫μ(t)g(t) dt + c. 5. Solve for y.

Example Solve the initial value problem ty' + 2y = 4t2, y(1) = 2. 1. Standard form: y' + (2/t)y = 4t. 2. Integrating factor μ(t) = e∫2/tdt =t2 3. Multiply the equation by μ(t) and write it in the form [t2y]' = 4t3 4. Integrate and obtain t2y = t4+ c. 5. Solve for y. y = t2 + 1/t2 , t > 0

Integral Curves for the initial value problem ty' + 2y = 4t2

1.3 Numerical Approximations: Euler’s Method When drawing direction field, many tangent line segments at successive values of t almost touch each other. By linking these an approximation to a solution of the differential equation can be obtained. 1. Can we carry out this linking of tangent lines in a simple and systematic manner? 2. If so, does the resulting piecewise linear function provide an approximation to an actual solution of the differential equation? 3. If so, can we say anything about the accuracy of the approximation? It turns out that the answer to each question is affirmative.

Euler Formula For the initial value problem y = f (t, y), y(t0) = y0, the Euler method is the numerical approximation algorithm yn+1 = yn + f (tn, yn)(tn+1 – tn), n = 0, 1, 2, . . . If uniform step size h is used. tn+1 = tn + h, yn+1 = yn + h f (tn, yn), To use Euler’s method, you simply evaluate the equations repeatedly.

Euler Formula (Ctd.) The result at each step is used to execute the next step. Thereby generate a sequence of values y1, y2, y3, . . . that approximate the values of the solution φ(t) at the points t1, t2, t3, . . . . Instead of a sequence of points, to get a function to approximate the solution φ(t), we use the piecewise linear function constructed from the collection of tangent line segments. That is, let y be given by y = y0 + f (t0, y0)(t − t0) in [t0, t1], and in general y is given by y = yn + f (tn, yn)(t − tn) in [tn, tn+1].

Example: Euler Method Problem: Use Euler’s method with a step size h = 0.2 to approximate the solution of the initial value problem dy/dt + 1/2y = 3/2− t, y(0) = 1 on the interval 0 ≤ t ≤ 1. Answer: To use Euler’s method, note that f (t, y) = −1/2y + 3/2 − t, so using the initial values t0 =0, y0 = 1, we have f0 = f (t0, y0) = f (0, 1) = −0.5 + 1.5 − 0 = 1.0.

Example: Euler Method (Ctd.) The tangent line approximation for t in [0, 0.2] is y = 1 + 1.0(t − 0) = 1 + t. Setting t = 0.2, we find the approximate value y1 of the solution at t = 0.2, namely, y1 = 1.2. At the next step we have f1 = f (t1, y1) = f (0.2, 1.2) = −0.6 + 1.5 − 0.2 = 0.7, and y = 1.2 + 0.7(t − 0.2) = 1.06 + 0.7t for t in [0.2, 0.4]. Evaluating at t = 0.4, we obtain y2 = 1.34. …

Example: Euler Method (Ctd.)

1. 4 Classification of Differential Equations: 1 1.4 Classification of Differential Equations: 1. Ordinary and Partial Differential Equations - If only ordinary derivatives appear in the differential equation then it is an ordinary differential equation. - If derivatives are partial derivatives then the equation is called a partial differential equation.

2.Systems of Differential Equations If there is a single unknown function, then one equation is sufficient. However, if there are two or more unknown functions, then a system of equations is required. For example, the Lotka–Volterra, or predator– prey, equations are important in ecological modeling. They have the form dx/dt = ax − αxy dy/dt = −cy + γ xy

3. Order The order of a differential equation is the order of the highest derivative that appears in the equation. The equations in the preceding sections are all first order equations. The equation ay'' + by' + cy = f (t), where a, b, and c are given constants, and f is a given function, is a second order equation.

4. Linear and Nonlinear Equations The ordinary differential equation F(t, y, y', . . . , y(n)) = 0 is said to be linear if F is a linear function of the variables y, y', . . . , y(n).Thus the general linear ordinary differential equation of order n is a0(t)y(n)+ a1(t)y(n-1)+… +an(t)y = g(t). An equation that is not of this form is a nonlinear equation.

φ(n)(t) = f [t, φ(t), φ'(t), . . . , φ(n-1)(t)] Some Terminology The process of approximating a nonlinear equation by a linear one is called linearization. A solution to y(n) = f (t, y, y', y'', . . . , y(n-1)) on the interval α < t < β is a function φ such that φ', φ'', . . . , φ(n) exist and satisfy φ(n)(t) = f [t, φ(t), φ'(t), . . . , φ(n-1)(t)] for every t in α < t < β.

Some Important Questions How can we tell whether some particular equation has a solution? This is the question of existence of a solution How many solutions it has, and what additional conditions must be specified to single out a particular solution? This is the question of uniqueness. Can we actually determine a solution, and if so, how?

Computer Use in Differential Equations A computer can be an extremely valuable tool in the study of differential equations. There exists extremely powerful and general software packages that can perform a wide variety of mathematical operations. Among these are Maple, Mathematica, and MATLAB.

CHAPTER SUMMARY Differential equations are used to model systems that change continuously in time. The mathematical investigation of a differential equation usually involves one or more of three general approaches: geometrical, analytical, and numerical.

1.1 Geometrical Approach A solution of dy/dt = f (t, y) is a differentiable function y = φ(t) such that φ'(t) = f (t, φ(t)). Hence f (t, y) specifies the slope of a solution passing through the point y at time t. This information can be displayed geometrically by drawing a line segment with slope f (t, y) at the point (t, y). A direction field (or Slope filed) consists of a large number of such line segments, usually drawn on a rectangular grid. A direction field provides a geometric representation of the flow of solutions of the differential equation.

1.2 Linear Equations – Analytical Approach dy/dt + p(t)y = g(t).Multiplying by the integrating factor μ(t)=exp[∫p(t)dt] leads to d(μy)/dt=μg. Integrating then yields y = μ−1[∫μg dt + c]. For the initial value problem y = f (t, y), y(t0) = y0, the Euler method is the numerical approximation algorithm tn+1 = tn + h, yn+1 = yn + h f (tn, yn), n = 0, 1, 2, . . . that geometrically consists of a finite number of connected line segments, each of whose slopes is determined by the slope at the initial point of each segment. 1.3 Numerical Approach

1.4 Classification of Differential Equations Differential equations are classified according to various criteria for a sensible treatment of theory, solution methods, and solution behavior: ordinary differential equations and partial differential equations single (or scalar) equations and systems of equations linear equations and nonlinear equations first order equations and higher order equations