Engineering Mathematics I

Transcription

1 Engineering Mathematics I_ 017 Engineering Mathematics I 1. Introduction to Differential Equations Dr. Rami Zakaria Terminolog Differential Equation Ordinar Differential Equations Partial Differential Equations Initial-Value Problem Nomenclature DE ODE PDE IVP Ref [1] Advanced Engineering Mathematics, 4th ed., b Dennis G. Zill & Warren S. Wright

2 Part 1: A review of the concept of differentiation

3 The concept of differentiation Suppose we have a variable that is a function of another variable, such as = f (). the rate Δ/ Δ eplains how fast changes for a certain shift in the value of. For a simple function such as =5, if changes from 1 to 4 (for eample), moves from 5 to 0. Therefore, the rate Δ/ Δ = 5 alwas!

4 Remember the definition of the derivative of a function: f ( ) lim f ( ) f ( ) 0 0 lim d it is not practical to appl the previous relation ever time we se ek to calculate the derivative of a function. This is wh we gath ered rules that we can appl directl according to the tpe of the function (see the attached table DERIVATIVES AND INTEGRALS ). Note: in this class we will use the (prime notion):,,,,..., n d d d d, and the derivative shape:,,,..., n (4) ( n) 4 ( Although we will avoid using the dot notion, ou should be aware of it)

5 The graphical meaning of differentiation The first derivative f of a curve f at a certain point is the tangent line on the curve at that point. f ) ( 0 m L 5 The first derivative f can be estimated from the slope of the tangent line at each point of f.

6 Dependent & Independent Variables When we write = f (), we are impling that reacts to t he changes of. In other words, changes first then foll ows. In this case, we call an independent variable, and a dependent variable. If we have T(,t) or T(t,), then T is dependent on both t a nd. In models, we can understand the independent variables as the inputs, while the dependent variables as the output. 6

7 Ordinar and Partial Derivatives Let f () be a function dependent onl on. We can then d calculate the ordinar derivative f () If we have g(,t) a function dependent on both and t, then it is possible to calculate the partial derivatives: f (, t) and f (, t) t 7 When we calculate a partial derivative we assume that one independent variable is changing while the others are constants.

8 8? ), ( Ordinar Derivative (Total Derivative!) ), ( Derivative Partial ), ( Derivative Partial ) ( Ordinar Derivative ) ( Ordinar Derivative ), (, ) ( 1, ) ( dt df f f e t dt d t t dt d f e t t t t t Eample - we have the following functions:

9 Part : Definition and Classification of A Differential Equation 9

10 What is a differential equation? A differential equation (DE) is an equation that contains the derivatives of one or more dependent variables, with respect to one or more independent variables. For eample: d is a differential equation where { }is the independent variable and { () } is the dependent variable A derivative of a function f() is another function f ()=d/, that can be found using the general differentiation rules. For eample : 1 e 5 6 Or: d e 6 e 10 If [] is m equation, then I can sa that [1] is a solution for that equation.

11 Another eample; If I have this differential equation: Then can ou tell if 4e 0.5 is a solution for it? The answer is es! It is a solution. In fact, this is not the onl solution. 0.5 Tr for eample ou will find it is also a solution. 6e But the big question is: HOW CAN I FIND THESE SOLUTIONS? We will answer this question during this course, but before that, we need to classif differential equations b tpe, order, and linearit. 11

12 Classification of Differential Equations Tpe ODE : Ordinar Differential Equation Contains derivates of onl one independent variable. Eamples: d sin d d e dt dt Order The order of a differential equation is the order of the highest derivative. Eamples: d d 0 e 4 order Linearit Linear Or Non-linear PDE : Partial Differential Equation Contains derivates of more than one independent variable. Eamples: t u u u t 1 0 The degree of a differential equation is power of the highest order derivative term. Eample: d d d d Degree 5 0 a 0 degree 1

13 Linear Differential Equation A differential equation is linear, if : 1. dependent variable and its derivatives are of degree one,. coefficients of a term do not depend upon dependent variable. Eamples: 1. d d 9 0. is linear.. 4 d d 6 is non - linear because in nd term is not of degree one. d d. is non - linear because in nd term coefficient depends on. 1

14 n th order linear differential equation 1. n th order linear differential equation with constant coefficients. a n d n n a n1 d 1... n1 a d a d n 1 0 a g. n th order linear differential equation with variable coefficients a n d n1 d 1... n a a a a g d d n

15 Solution of an ODE A function ϕ() is a solution of an ODE on an interval (domain) I, if it satisfies the ODE on I. For Eample : =+c 1 is a solution of a 1 st order ODE On the interval (-, ) d Eercise (P5) : Verif that : is a solution of the DE: 0.5 On the interval (-, ) Solution: Left-hand side: right-hand side: Notice in this eample =0 is one of the solutions. In DEs we call the zero solution a trivial solution.

16 Eplicit and Implicit solutions An Eplicit Solution is a solution where the dependent variable is epressed in terms of the independent variable and constants. In other words, it is a solution written as =ϕ(). Sometimes, when we solve a DE, we don t directl get an eplicit solution. Instead we find a function G(,)=0 that satisfies the equation; we call it an Implicit Solution. Eample : 5 is an implicit solution of the differential equation d In fact an function c would satisf the equation; where c is an arbitrar constant 16

17 Another eample: 4 9 c is a solution for the DE: Notice that the solution is a famil of ellipses. Observe that at an given point ( 0, 0 ), there is a particular solution (unique) curve of the above equation which goes through the given point. We call this tpe of solution a Famil of solutions or a General Solution * Remember that an equation can also have a singular solution that is not a member of a famil! A Sstem of Differential Equations: Until now we are discussing one differential equation. But in man cases we will need to solve two or more equations that share the same independent variable together. For eample, when we look for a solution of this sstem of equations: f ( t,, ) dt d g( t,, ) dt we are actuall looking for solutions =ϕ1(t) and = ϕ(t) that satisf the two equations at the same time. 17 * Usuall we use the term General Solution if the famil of solutions are the onl solutions of the equation (which is true in man cases).

18 Eercises: 1. Determine the order and linearit of the following ODE: Equation Order Linearit d d 0 Non-linear t u (4) uu t 0 sin e e

19 5. Verif that : c1e ce is a general solution (famil of solutions) for the ODE: 0 6. Verif that : is a solution for : 0 on the interval (-, ) 0 19 See more eamples at P.10

20 . Verif that : / e is a solution of the ODE: 0. Verif that : e cos is a solution of the ODE: See more eamples at P.9

21 4. Verif that : 1 is an implicit solution of the ODE: ( ) d 0 and then find the eplicit solution(s). 1

22 6. Determine if : 5 9 has (accepts) a constant solution or not. Assume: Then: c c 9 c 0 is a solution 7. Determine if : 1 1 has (accepts) a constant solution or not. 8. Determine if pair of functions: e t is a solution for the sstem of differential equations: 6t t e, e 6t 5e d, 5 dt dt