These notes are based mostly on [3]. They also rely on [2] and [1], though to a lesser extent.

Transcription

1 Chapter 1 Introduction These notes are based mostly on [3]. They also rely on [2] and [1], though to a lesser extent. 1.1 Definitions and Terminology Background and Definitions The words "differential equations" suggest we will be working with equations. An equation is the statement that two mathematical expressions are equal. The equations studied in an algebra class were of the form x 2 + 5x + 3 = 0 or 2x 5 = 1. The unknown, denoted by the variable x in our examples, represents a number. Solving such equations means finding the value or values of x which make the equation a true statement. The equations we will study in these notes will be different. The word "differential" suggest they will contain derivatives. Let s look at a simple example. Example Let y denote some function of x. When we want to emphasize that y is a function of x, we will write y (x). If we let y = e 3x2 then y = 6xe 3x2 = 6xy. In other words, y satisfies the equation y = 6xy. This is a diff erential equation. Remark We quickly review some elementary notions and notation from Calculus, then we will define what a differential equation is. Let y = y (x) (some function of x). 1. x is called the independent variable, y is called the dependent variable. 2. When y depends only on one variable x, the derivative of y (with respect to x) is denoted y dy. It can also be denoted using Leibniz notation: dx. This notation is more explicit as it specifies clearly that it is the derivative 1

2 2 CHAPTER 1. INTRODUCTION of y with respect to x. The variable on top is the dependent variable, the variable at the bottom is the independent variable. 3. Higher order derivatives are denoted y or d2 y dx 2, y or d3 y. For higher dx3 orders, the notation is y (4) or d4 y dx 4, and in general y(n) or dn y for n 4. dxn 4. For functions of several variables, the derivatives are called partial derivatives. For example, if u is a function of x and y then the partial derivatives are denoted u x, u. This is similar to Leibniz notation. y 5. The derivative dy represents the instantaneous rate of change of y with dx respect to x. Students should remember from calculus that if y gives the position of an object, then the derivative dy represents the velocity of the dx d 2 y object and the second derivative, dx 2 represents the acceleration of the object. 6. When we say that a quantity A is proportional to another quantity B, we mean that there exists a constant k such that A = kb. Example The third order derivative of y = y (x) with respect to x is denoted d3 y dx 3. Example The fourth order partial derivative of u = u (x, y, t) with respect to t is denoted 4 u t 4. Example Let P = P (t) denote the population of a city as a function of time and assume P (t) > 0 for every t. What equation corresponds to the statement "P increases at a rate proportional to P ". The rate of increase of P is the same as the rate of change and is therefore dp dp. We are simply saying that constant k such that dp are talking about the rate of increase of P, it means dp have k > 0 since P > 0. is proportional to P hence, there exists a = kp. We can actually say more about k. Since we > 0 hence, we must Remark In the above example, if we had said "P decreases at a rate proportional to P ", then the corresponding equation would have been dp = kp where k < 0 (why?). Note that we can also write dp = kp where k > 0. The latter equation is the one we will use.

3 1.1. DEFINITIONS AND TERMINOLOGY 3 Example Let T (t) represent the temperature of an object placed in a room and T e denote the temperature of the room. We will assume T e does not change. If T e is different from the temperature of the object, then the temperature of the object will change with time. It will increase if the room is warmer than the object and decrease otherwise. Newton s law of cooling states that the temperature of the object will increase/decrease at a rate proportional to the diff erence between the temperature of the object and the temperature of the room. Write an equation corresponding to Newton s law of cooling. The rate of change of the temperature of the object is dt, so Newton s law of cooling says that dt = k (T T e) where k is a positive constant. We can verify that this equation agrees with our intuition. If the temperature of the object is less that that of the room, then T < T e hence k (T T e ) < 0 thus dt < 0 which means that T will decrease, which agrees with what we would observe. We are now ready to define differential equations. Definition A diff erential equation (DE) is an equation containing one or more unknown functions and some of their derivatives. When the unknown function is a function of one variable, the corresponding diff erential equation is called an ordinary diff erential equation or ODE. When the unknown function is a function of several variables, the corresponding diff erential equation is called a partial diff erential equation or PDE. We will only discuss ODEs in these notes. Example dy dx + 3x d2 y dx 2 + sin x = 0 This is an ODE. Example y + sin y + y = cos x This is an ODE. Example y = k where k is a constant. This is an ODE. Example u t = 2 u x u where u = u (x, y, t) y2 This is a PDE Where do ODEs Come From and What Do We Do With Them? We begin with some general remarks about differential equations. Differential equations arise from mathematical models which are developed in the sciences to help understand physical phenomena.

4 4 CHAPTER 1. INTRODUCTION When the model involves only one unknown quantity, we only need one differential equation. When the model involves several unknown quantities, we will need as many equations as there are unknown quantities. Sometimes, we can solve the differential equations. In this class, we will focus on solving differential equations (quantitative study). Often, we do not know how to solve the differential equations. However, we can usually still get information about the solution such as whether the unknown quantities are increasing, decreasing, going to infinity, going to 0, reaching an equilibrium,... (qualitative study). We now look at some examples. Example Consider the free fall of an object where an object is released from a certain height above the ground and subject only to gravity. Newton s second law of motion tells us that mass times acceleration equal the total force acting on the object. If we let h = h (t) be the height above the ground of the object and m be the mass of the object, then the acceleration of the object is d2 h 2 and the force due to gravity is mg so we obtain m d2 h 2 = mg or d 2 h 2 = g This is a diff erential equation, h (t) is the unknown function. Fortunately, this one is easy to solve. Integrating once with respect to t gives Integrating again gives dh = gt + C 1 h (t) = 1 2 gt2 + C 1 t + C 2 We will see that the constants C 1 and C 2 can be determined from the initial height and velocity of the object. Example Radioactive decay is another classical example from calculus. The model for radioactive decays says that the mass m of a radioactive object decreases at a rate proportional to the mass of the object. Derive a diff erential equation from this statement, then solve the equation to find a formula for m as a function of time. As we saw above, the equation we obtain is dm = km

5 1.1. DEFINITIONS AND TERMINOLOGY 5 Which can be written as Integrating on both sides gives dm m = k ln m = kt + C or m = e kt+c = m 0 e kt where m 0 = e C Classification of ODEs Even when we can solves differential equations, finding a solution is usually quite diffi cult. There is not one technique which works for all the equations. We classify differential equations into different categories. These categories will then determine how the equations are solved. The categories we will study in this class are: Type: If all the derivatives which appear in an equation are with respect to a single independent variable, we have an ordinary differential equation (ODE). Otherwise, we have a partial differential equation (PDE). We ll focus on ODEs in this class. Order: The order of a differential equation is the order of the highest order derivative present in the equation. Linearity: A differential equation is said to be linear if it is linear in its dependent variables and its derivatives that is if its dependent variable and its derivatives appear in additive combinations of their first power. In other words, a linear ODE is of the form a n (x) dn y dx n + a n 1 (x) dn 1 y dx n a 1 (x) dy dx + a 0 (x) y = b (x) where a 0 (x),..., a n (x) and b (x) only depend on x, the independent variable and some of these functions could be 0. We will only discuss linearity for ODEs. Example For each equation below, find the independent and dependent variables, then determine its type, its order and if it is linear. 1. d 3 y dy + sin x dx3 dx = ex

6 6 CHAPTER 1. INTRODUCTION Type: The dependent variable is y, there is only one independent variable, x. Hence, this is an ODE. Order: 3 Linear? yes, it is of the form a 3 (x) d3 y dx 3 + a 2 (x) d2 y dx 2 + a 1 (x) dy dx + a 0 (x) y = b (x) with a 3 (x) = 1, a 2 (x) = 0, a 1 (x) = sin x, a 0 (x) = 0 and b (x) = e x. 2. d 4 y dy + sin x dx4 dx = ey Type: The dependent variable is y, there is only one independent variable, x. Hence, this is an ODE. Order: 4 Linear? no, because of e y. ( 3. (1 y) x + dy ) = sin x dx Type: The dependent variable is y, there is only one independent variable, x. Hence, this is an ODE. Order: 1 Linear? no, because of y dy dx 4. 2 u x u y 2 = ex sin y Type: The dependent variable is u, the independent variables are x and y. Hence, this is a PDE. Order: 2 Linear? we only discuss linearity of ODEs. 5. t d2 x 2 + xdx = sin t Type: The dependent variable is x, there is only one independent variable, t. Hence, this is an ODE. Order: 2 Linear? no, because of x dx

7 1.1. DEFINITIONS AND TERMINOLOGY Exercises 1. Do # 1, 3, 5, 7, 9, 11, 13, 15 at the end of section 1.1 in your book. 2. Write the general form of a linear ordinary differential equation of order Write the general form of a linear ordinary differential equation of order 3.

8 Bibliography [1] Paul Blanchard, Robert L. Devaney, and Glen R. Hall, Diff erential equations, fourth ed., Brooks/Cole, CENGACE Learning, 2012 (English). [2] Charles H. Edwards, David E. Penney, and David T. Calvis, Diff erential equations and boundary value problems: Computing and modeling, fifth ed., Pearson, 2015 (English). [3] R. K. Nagle, Edward B. Saff, and Arthur D. Snider, Fundamentals of differential equations, eigth ed., Pearson/Addison-Wesley, 2012 (English). 185