Introduction to Differential Equations

Transcription

1 CHAPTER 1 Introduction to Differential Equations Differential equations arise from real-world proble ms and problems in applied mathematics. One of the fi rst things you are taught in calculus is that the derivative of a function is the instantaneous rate of change of the fu nction w ith respect ro its independent variable. When mathematics is applied to real-world problenns, it is often the case that finding a relation between a function and its rate of change is easier than finding a fommla for the function itself; it is thi s relation between an unknown function and its derivatives that produces a differential equation. To give a very si mple example, a biologist studyi ng the growth of a population, with size at time r given by the function P(t), might make the ve1y simple, but logical, assumption that a population grows at a rate directly proportional to its size. ln mathematical notation, the equation for P(r) could then be written as dp - = rp(t). di where the constant of proporti onality, r, would probably be determined experimentally by biologists working in the field. Equations U1sed for modeling population growth can be much more complicated than this, sometimes involving scores of interacting populations with different properties; however, almost any population model is based on equations similar to thi s. In an analogous manner, a physicist might.argue that all the forces acting on a particular moving body at ti me / depend only on its position x(i) and its velocity x'(r). He could then use Newton's second law to express mas.s times acceleration as mx" (t ) and write an equation for x(1) in the fo m1 mx"(1) = F(x(1), x 1 (1)). where F is some function of two vari ables. One of the best-known equations of this type is the spring-mass equation mx" +bx' + kx = f(r), ( I. I) in which x(r) is the position at ti me r of an object of mass m sm-pended on a spring, and band k are the damping coefficient and s pring constant, respectively. The function f represents an external force acting on the system. Notice that in (1. 1 ), where x is a function

2 2 1. Introduction to Differential Equations of a single variable, we have used the convention of omitting the independent variable 1, and have written x, x', and x" for x(t) and its derivatives. Jn both of the examples, the problem has been written in the fonn of a di fferential equation, and the solution of the problem lies in finding a function P(t), or x(t), which makes the equation true Basic Terminology Before beginni ng to tackle tl1e problem of formul ating and solving di fferential equations, it is necessa1y to understand some basic tem1inology. Our first and most fundamental definition is that of a differential equation itself. Definition I.I. A differential equation is any equation involving an unknown function and one or more of its derivatives. I. TI1e following are examples of differential equations: P 1 (t) = rp(t)(i - P(t)/N)- H harvested population growth drr d' x + 09dx. Tr +2x = 0! "(1) + 4/(1) = sin(w1) y' 1 (1) + µ(y 2 (r ) - l )y 1 (1) + y(1) = 0 ;j2 J2 Jx'Tll(X. y) + JY'Ill(X. y) = 0 spring-mass equation RLC circuit showing "beats" van der Pol equation Laplace's equation A \\\1\\1\ \\\\\\\\\ '\ '\ H '>< Y.I'<. '\' : \\\\'\\\'\) For the first four equations the graphs abo'e illustrate different ways of picturing the solution curves Ordinary vs. Partial Differential Equations Differential equations fall into two veiy broad categories, called ordinary di fferential equations and partial differential equations. If the unknown function in the equation is a function of only one variable, the equation is called an ordinary differential equation. ln the list of ex amples, equations l-4 are ordinary dj fferential equations, with the unknown functions being P(t ), x(t ), / (1), and y(r) respectively. lf the unknown function in the equation depends on more than one independent variable, the equation is called a partial different ial equation, and in this ca.>;e, the derivat'.ives appeari ng in the equation wi ll be partial derivatives. Equation 5 is an example of an umportant partial differential equation, called Laplace's equation, which arises in several areas of applied mathematics. In equation 5, u is a function of the two independent variable.s x and y. In this book, we will not consider

3 1.1. Basic Terminology 3 methods for solving partial differenti al equations. One of the basic methods involves reducing the partial differential equation to the solution of two or more ordinary di fferential equations, so it is important to have a solid g rounding in ordinary di fferential equati ons first Independent Variables, Oependent Variables, and Parameters Three different types of quantities can appear in a differential equation. TI1e unknown function, for which the equation is to be sohoed, is called the dependent variable, and since we w ill be considering only ordinary differential equations, the dependent variable is a function of a single independent ''ariable. [n addition to d1e independent and dependent variables, a third type of variable, called a pnrameter, may appear in the equation. A parameter is a quantity that remains fixed in any specification of the problem, but can vary from problem to problem. In this book, parameters will usually be real numbers, such as '" N, and H in equation 1, tu in equation 3, andµ in equation Order of a Differential Equation Another important way in which differential equations are classified is in tem1s of their order. Definition 1.2. The order of a differential e quation is the order o f the highest derivative of the unknown function that appears in the equation. The differential equation I is a first-order equation and the others are al l second-order. Even though equati on 5 is a partial di fferential equation, it is still said to be of second order since no de rivatives (in this case partial derivatives) of order higher than two appear in the equation. You may have noticed in the Table of Contents that some of the chapter headings refer to first-order or second-order di fferential equations. In some sense, first-order equations are thought of as being simpler than second-order equations. By the time you have worked through Chapter 2, you may not want to believe that this is true, and there are special cases where it defi nitely is not true; however, it is a useful way to distinguish between equati ons to which different methods of solution apply. In Chapter 4, we will see that solving ordinary differential equations of order greater than one can always be reduced to solving a system of first-order equations What is a Solution? Given a differential equation, exactly what d!o we mean by a solution? h is fi rst important to real ize that we are looking for a function, and therefore it needs to be defined on some interval of its independent variable. Before computers were available, a solution of a differential equation usually referred to an a nalytic solution; that is, a fonnula obtained by algebraic methods or other methods of mathematical analysis such a~ integration and differentiation, from which exact values of the unknown function could be obtained. Definition 1.3. An analytic solution of a differential equation is a sufficiently differentiable function that, if substituted into the equation, together with the necessary derivatives,

4 4 1. Introduction to Different ial Equations makes the equation an identity (a true statement for al l values of the independent variable) over some interval of the independent variab le. It is now possible, however, using sophisticated computer packages, to numerically approximate solutions to a differential equation to any desired degree of accuracy, even if no formul a for the solution can be found. You w ill be introduced to numerical methods in Chapter2, and many of the equations in later chapters will only be solvable using numerical or graphical methods. Given an analytic solution, it is usually fairly easy to check whether or not it satisfies the equation. ln Examples I. I. I and I. 1.2 a formula for the solution is ghen and you are only asked to veri fy that it satisfies the given differential equation. Example Show that che function p(t) = e- 21 is a solution of che di fferential equation x" + 3x 1 + 2" = 0. Solution To show that ic is a solution, compute the first and second derivatives of p(1): p 1 (1) = -2e- 21 p" (r) = 4e- 2 '. With the three functions p(1), p'(1), and p"(t) substituted into the differential equation in place of x, x', and x", ic becomes which is an ide ntity (in the independent varia ble 1) for all real values of 1. When showing that both sides of an equation are identical for al I values of the variables, we wi I I use the equivalence sign = This wi IL be used as a convention throughouc the book. For practice, show that the function q(1) = Je- 1 is also a solution of the equation x" + 3x' + 2x = 0. le may seem surprising t hat two completely different functions satisfy this equation, but we will soon see that differential equations can have many solutions, in fact infi nitely man)'. In the above ex runple, the solutions p and q turned out to be functions that are defi ned for all real values of 1. tn the next example, things are not quite as simple. Example Show that the function (1) = ( I ) = v1l=t2 is a solucion of thedifferentialequation x' = -1/x. Sofwion First, notice that (1) is not even defined oucside of che interval -I :;: 1 :;: l. In the interval - I < 1 < I, (1) cru1 be differentiated by the chain ru le (for powers of functions): t/1'(1)= ( 1/ 2)(1-1 2 ) (-21) = -1/(1-1 2 ) The 1ight-hand side of the equationx' = -r /x, with (1) substituted for x, is -t/ t/1(1) = -1/(J -,2)112, which is identically equal to t/1'(1) wherever</> and ' are both defined. Therefore, (1) is a solution of the differential equation x' = -1/x on the interval (-1. I).

5 1. 1. Basic Terminology 5 You may be wondering if there are any solutions of x' = -I / x that ex ist outside of the interval - I < r < I, since the dj fferentiail equation is certainly defined outside of that interval. This problem wi ll be revisited in Section 2.4 when we study the ex istence and uniqueness of solutions, and it wi ll be shown that solutions do exist throughout the entire (1, x )-plane Systems of Differential Equations In Chapter 4 we are going to study systems <>f differential equations, where two or more dependent variables are related to each other by differential equations. Linked equations of this sort appear in many real-world appl ications. As an example, ecologists studying the interaction between competing species in a particular ecosystem may find that the growth (think derivative, or rate of change) of each population can depend on the size of some or all of the other populations. To show that a set of formulas for the unknown populations is a solution of a system of this type, it must be shown that the functions, together with their derivatives, make every equation in the system an identity. The fo llowing simple example shows how this is done. Example Show that the functions x (r) = e- 1, y(r) = -4e- 1 form a solution of the system of dj fferential equati ons x 1 (1) = 3x + y y'(1) = -4x - 2y. (1.2) So/utio11 The derivatives that we need are x' (t) = -e- 1 and y'(t) = -(-4e-') = 4e-'. Then substitution into ( 1.2) gives 3x + y = 3(e- 1 ) + (-4e- 1 ) = (3-4)e- 1 = -e- 1 = x'(t), -4x - 2y = - 4(e- 1 )-2(-4e- 1 ) = ( )e- 1 = 4e- 1 = y'(1); therefore, the given functions for x and y fom1 a solution for the system. Exerd ses 1.1. For each equation 1-8 below, determine its order. Name the independem variable, the dependent variable, and any parameters in the equation. l.dy/dr= y 2 -I 2. d P/ dr = rp(i - P/ k) 3. dp/ dr = rp(i- P/k)- ff PP\ a-+ 4. mx" +bx' + kx = 21 5, assuming x is a function of r 5. x"' + 2, " + x' + 3x = sin(wr). assumj ngx is a fu nction of r 6. (ty' (1))' = ae 1 7. d 2 9/ dt 2 + sin(9) = 4cos(1)

6 6 1. Introduction to Differential Equations 8. y" + s(y 2 - l)y' + y = 0, assuming y is a fu nction of / For each equation 9-17 below, show that the given function is a solution. Determine the largest intel>'al or intervals of the independent vari able over which the solution is defined, and satisfies the equation. 9. 2Y" + 6x' + 4x = 0. x(1) = e x" + 4x = 0. x(1) = s in(21) + cos(21) x " + 31x' + x = 0. x(t) = l/ x " + 31x' + x = 0, x(i) = ln(1)/ P' = rp, P(1) =Ce", C any real number 14. P' = rp( I - P). P (1) = 1/ ( 1 + ce-r 1 ), C any real number 15. x' = (1 + 2)/ x, x(1) = -J I 16. x " - 21x' + 6x = 0, x(1) = r y" + 1y' + (1 2 - ±)Y = 0. y(1) = "'? In the nex t four problems, show that the given functions form a solution of the system. Detenn ine the largest intel>'al of the independent variable over which the solution is defined, and satisfies the equations. 18. System: x' = x - y. y' = - 4x + y, solution is x = e- 1 -e 31. y = 2e- 1 +2e System: x' = x - y. y' = 4x + y. wlution is x = e 1 cos21 - te 1 sin 21, y = e 1 cos21 + 2e 1 sin System: x' = y, y' = - I Ox - 2y, solution is x = e- 1 sin(31), y - e- 1 (3 cos(31) - sin(3r)). 21. System: x' = x + 3y, y' - 4x + 2y, solution is x - 3es 1 + e-2 1, y _ 4es' _ e Families of Solutions, Initial-value Problems In this section the solutions of some vel)' simple differential equations will be examined in order to give you an understanding of the tenns 11-parameter family of solutions and general solution of a differential equation. You will also be shown how to use certain types of infom1ation to pick one particular solution out of a set of solutions. While you do not yet have any formal mefihods for solving differential equations, there are some \ery simple equations that can be solved by inspection. One of t11ese is.x' =.. -r. (1.3) This first-order di fferential equation asks you to find a function x(1) which is equal to its own derivative al evel)' value of t. Any calculus student knows one function that satisfies this property, namely the exponential functio:n x(1) = e 1 In fact, one reason mathematicians use e as the basis of their exponential function is that e 1 is the only function of the