! 1.1 Definitions and Terminology

Transcription

1 ! 1.1 Definitions and Terminology 1. Introduction: At times, mathematics aims to describe a physical phenomenon (take the population of bacteria in a petri dish for example). We want to find a function that will output the number bacteria, say y, at any given time t. We seek to find y = P( t). Sometimes, finding that function y is impossible, or possibly it is much easier to describe/ measure the rate of change of the population. Consider the statement, the population is growing at a rate which is proportional to its size. We can easily convert that statement into mathematical symbols: This is an example of a differential equation. Loosely speaking, it is an equation involving derivatives. 2. Example 1: Your friend hands you the equation. They ask, who is the equation y = 3xy that satisfies this equation? Answering this, and far more complex differential equations, is the focus of the course. Note: Many very awesome real world applications lend themselves readily to being described by differential equations. They are not the main focus of the course, our focus will be on solving differential equations. 3. Definition 1.1.1: Differential Equations: An equation containing the derivatives of one or more unknown functions (or dependent variables), with respect to one or more independent variables, is said to be a differential equation (DE). 4. Classification by Type: Ordinary Differential Equation (ODE): This is a differential equation with only ordinary (as in the 5A/5B sense) derivatives of one or more unknown functions with respect to a single variable. (These are our focus) Partial Differential Equation (PDE): This is a differential equation involving partial derivatives of one or more unknown functions of two or more independent variables. Section 1.1 Page 1

2 Example 2: Identify each equation as an ODE or a PDE: ( a) + 8y = t 5 ( b) u y = v x ( c) du + dv = 2u + v ( d) 2 u x + 2 u 2 y = 0 2 ( e) y = e2t 5. Notation! Leibniz notation: This is the far superior notation, we see immediately the independent variable(s) and the dependent variable. Prime notion: This notation is more compact and usually easier to look at, but it loses the dependent variable. So we have to be careful and always keep track of who we are taking derivatives with respect to. Newton s dot notation: I will not use this, but will remind you of it. d = y' = yi 2 y d = y'' = yii 2 3 y d = y''' = yiii 3 n y n = y n Example 3: We can rewrite Example 2 part (e) in prime notation. (Note the loss of context of t) y = 2 e2t 6. Classification by Order: The order of a differential equation is the order of the highest derivative in the equation. For example: y = ex is a -order ODE. Section 1.1 Page 2

3 (a) Normal Form: Consider the ODE. This equation is equivalent to + 5 8y = 2 e2t or. We can think of the LHS as a function of + 5 8y 2 e2t = 0 y''+ 5y' 8y e 2t = 0 the 4 variables t, y, y' and y''. In general, we can write any nth order ODE in the form = 0 F t, y, y',..., y ( n) (F is a function of variables). We assume now that we can solve for y n uniquely, in terms of the variables. The differential equation becomes (where f is a real-valued continuous function). We call this form the normal form. Example 4: Write each equation in normal form. (Note the order of each equation) ( a) xy'+ 8y = x 5 b y''+ 2y' xy = 0 (b) M ( x, y) + N ( x, y) = 0 Form: For first-order ODEs M ( x, y) + N ( x, y) = 0 By example, consider 3x 2 y'+ e 2 x y = 5x : ( ) = 0 7. Classification by Linearity: An nth order ODE F x, y, y',..., y n is said to be linear if it is linear in y, y',..., y n. Concrete example: Generally, we write the linear nth order ordinary differential equation as: a n ( x) d n y + a ( x) d n 1 y n n 1 +!+ a ( x) n a 0 ( x)y = g( x) or ( x)y ( n) + a n 1 ( x)y ( n 1) +!+ a 1 ( x)y'+ a 0 ( x)y = g( x) a n where we recognize a n ( x), a n 1 ( x),..., a 1 ( x), a 0 ( x) as functions of the one independent variable x. In other words, the function y and all of its derivatives are to the first power. Section 1.1 Page 3

4 First-order linear ODE (in general): Second order linear ODE (in general): Example 5: Determine whether each equation is linear or non-linear. State its order. 7 (a) cost d 3 t (b) du du + t = u 2 cosu d 3 t du du + t = u2 8. Solutions: Definition 1.1.2: Solution of an ODE Any function φ, defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into an nth-order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval. (a) Sometimes we will denote the solution y x. (b) Given y = e 3x, then. Written another way,. = = Thus, the solution to the linear ODE = 3y is. (c) The interval I referred in is sometimes called the interval of definition, the interval of existence, the interval of validity, or the domain of the solution. It can be of the form ( a, b), [ a, b), ( a, ), etc. [It is a portion of the real number line!] It is the part of the number line for which the solution function satisfies the DE. If you solve a DE, you must think about the interval of existence. (d) Example 6: Verify the indicated function is a solution of the given DE on the interval,. 1 (i) ; = xy 2 y = 1 16 x4 *Do you see another trivial solution to the DE in part (i) Section 1.1 Page 4

5 (ii) y" 2y'+ y = 0 ; y = xe x * A solution of a differential equation that is identically zero on an interval I is said to be a trivial solution. Is y = 0 a solution to (ii)? 9. Solution Curve: If we graph the solution φ (or φ( t) for example) of an ODE, we call this graph a solution curve. There is a difference between the graph of φ and the solution curve φ. Because φ is a solution of a DE, it is differentiable and thus continuous. In other words, there may be a difference between the interval of definition for the solution to the ODE and the domain of the function φ. Example 7: A solution to the ODE is. (Verify if you like.) But, y is not = 1 2xy2 y = 4 x 2 differentiable at x = 2 or x = 2, and is discontinuous at x = 2 or x = 2. A solution to a DE is an equation that is differentiable on and interval and satisfies the equation. So, a solution to is ON ANY INTERVAL that does not contain or = 1 2xy2 y = x = 2 4 x 2 ( 2, 2) x = 2. We will take I to be as large as possible. Here I would be either, 2,, or ( 2, ). More on this later. 10. Explicit and Implicit Solutions: Recall from calculus implicitly and explicitly defined functions ( y = x 2 + cos x versus x 2 + y 2 = e xy ). If our solution of an ODE has the dependent variable expressed only in terms of the independent variable, we have an explicit solution. All solutions from examples 6 and 7 (including y = 0 for all) were explicit solutions. We will be solving ODEs. Sometimes the best we can do is finding an implicit solution, which will be a relation or expression of the form G x, y that defines implicitly. = 0 φ Section 1.1 Page 5

6 Definition 1.1.3: Implicit Solutions of an ODE A relation G( x, y) = 0 is said to be an implicit solution of an ordinary differential equation on an interval I, provided that there exists at least one function φ that satisfies the relation as well as the differential equation on I. 4 Example 8: (a) Verify that the relation y 2 2x 2 y = 1 is a 3 solution of the ODE 2xy + ( x 2 y). = (b) Find at least one explicit solution φ to the ODE (be sure to include the interval of definition I ). 11. Families of Solutions: When and why in integral calculus do we encounter a constant of integration, often called c? Recall, this gave us a family of curves. Analogously, we will usually be encountering a constant or parameter c (or many of them) when solving ODEs. = 0 (a) When solving a first-order DE F x, y, y', we usually obtain a solution containing a single parameter c. If this is the case, the set of solutions = 0 y = ce x = 0 G x, y, c (don t let this notation scare you, an example of G x, y, c is ) is called a one-parameter family of solutions. Section 1.1 Page 6

7 (b) When solving an nth order DE = 0 F x, y, y',..., y ( n) = 0 family of solutions G x, y, c 1, c 2,, c n., we seek to find an nth parameter (c) Thus: a single ODE can have infinitely many solutions corresponding to different choices of parameter(s). c i (d) A solution to a differential equation that has NO parameters is called a particular solution. Example 9: Consider the linear second order differential equation y = 0 The (circle the correct choice) one / two -parameter family y = c 1 e 3x + c 2 xe 3x is an explicit / implicit solution to the ODE. (If you don t trust me, verify, but you will do something similar in #23). A is y = 2e 3x 4xe 3x corresponding to c 1 = 2, c 2 = 4. (Our selection of the c i s will depend on given criteria, more to come). Graphed to the right are some particular solutions with different selections of c i s. (e) Singular Solution: Sadly, sometimes a differential equation possesses a solution that is not a member of a family of solutions of the equation (i.e., no choice of c will produce said solution). By example, the family of solutions of y' = xy 1/2 is y = x2. No selection of will 4 + c c produce the trivial (and now singular) solution y = 0. Section 1.1 Page

8 12.Systems of Differential Equations: It is rather common that application problems lend themselves to systems of DEs. A system of ordinary differential equations is two or more equations involving the derivatives of two or more unknown functions of a single independent variable. For example = f ( x, y, t ) = f ( x, y, t ) = 5x 8y + et = 7y + 6x are examples of first order systems of ODEs, where x and y are the dependent variables of the independent variable t. y = φ 2 ( t) A solution of a system like above is a pair of functions x = φ 1 t and, defined on a common interval I, that satisfy each equation of the system on this interval. Section 1.1 Page 8