TOC 1 Introduction to Differential Equations 1.1 Preliminaries 1.2 - PDF document

TOC 1 Introduction to Differential Equations 1.1 Preliminaries 1.2 Differential Equations; Basic Terminology 1.3 n -Parameter Family of Solutions; General Solution 1.4 Initial Conditions and Initial-Value Problems 2 First Order Differential Equations and Appli- cations 3 Second Order Linear Differential Equations 4 The Laplace Transform 5 Linear Algebra 6 Systems of Linear Differential Equations Appendices: Appendix-1 Complex Numbers Appendix-2 Polynomials Appendix-3 Tables

DIFFERENTIAL EQUATIONS 1.1 Preliminaries See Section 1.1 in the textbook at CASA. • Real numbers and intervals • Functions • Limits and continuity Thm. 1: f continuous on a closed in- terval then . . . • Derivatives – Thm. 2: f differentiable at x then . . . – Thm. 3: Mean Value Theorem – Corollaries – ( f ± g ) ′ , ( fg ) ′ , ( f/g ) ′ , etc. 1

• Integration: � b � x � f ( x ) dx, a f ( x ) dx, F ( x ) := a f ( t ) dt, etc. – Thm. 4: integral vs. antiderivative – Thm. 5: Fundamental Thm. of Calcu- lus – Cor’s: f ′ = 0, f ′ = g ′ on an interval � f ± g , � f · g , � f/g , etc. – – Integration by parts, change of variable

1.2. Basic Terminology A differential equation is an equation that contains an unknown function to- gether with one or more of its deriva- tives. 2

Examples: y ′ = 2 x + cos x 1. dy 2. dt = ky (exponential growth/decay) x 2 y ′′ − 2 xy ′ + 2 y = 4 x 3 3. 3

∂ 2 u ∂x 2 + ∂ 2 u 4. ∂y 2 = 0 (Laplace’s eqn.) d 3 y dx 3 − 4 d 2 y dx 2 + 4 dy 5. dx = 0 4

TYPE: If the unknown function depends on a single independent variable, then the equation is an ordinary differential equation (ODE). If the unknown function depends on more than one independent variable, then the equation is a partial differential equation (PDE). 5

ORDER: The order of a differential equation is the order of the highest derivative of the unknown function appearing in the equation. 6

Examples: y ′ = 2 x + cos x 1. dy 2. dt = ky (exponential growth/decay) x 2 y ′′ − 2 xy ′ + 2 y = 4 x 3 3. 7

∂x 2 + ∂ 2 u ∂ 2 u 4. ∂y 2 = 0 (Laplace’s eqn.) d 3 y dx 3 − 4 d 2 y dx 2 + 4 dy 5. dx = 0 d 2 y +3 e xy = d 3 � dy � dx 3 ( e 2 x ) 6. dx 2 +2 x sin dx 8

SOLUTION: A solution of a differential equation is a function defined on some domain D such that the equation reduces to an identity when the function is substi- tuted into the equation. 9

Examples: y ′ = 2 x + cos x 1. 10

y ′ = ky 2. 11

y ′′ − 2 y ′ − 8 y = 4 e 2 x 3. y = 2 e 4 x − 1 2 e 2 x Is a solution? 12

y ′′ − 2 y ′ − 8 y = 4 e 2 x y = e − 2 x + 2 e 3 x Is a solution? 13

x 2 y ′′ − 4 xy ′ + 6 y = 3 x 4 4. 2 x 4 + 2 x 3 y = 3 Is a solution? 14

x 2 y ′′ − 4 xy ′ + 6 y = 3 x 4 y = 2 x 2 + x 3 Is a solution? 15

∂ 2 u ∂x 2 + ∂ 2 u 5. ∂y 2 = 0 x 2 + y 2 � u = ln Solution? 16

∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = 0 u = cos x sinh y , u = 3 x − 4 y Solutions?? 17

Finding solutions 6. Find a value of r , if possible, y = e rx such that is a solution of y ′′ − 3 y ′ − 10 y = 0 . 18

7. Find a value of r , if possible, y = x r such that is a solution of x 2 y ′′ + 2 x y ′ − 6 y = 0 . 19

8. Find a value of r , if possible, y = x r such that is a solution of y ′′ − 1 x y ′ − 3 x 2 y = 0 . 20

From now on, all differential equa- tions are ordinary differential equa- tions. 21

1.3. n -PARAMETER FAMILY OF SOLUTIONS / GENERAL SOLU- TION Example: Solve the differential equa- tion: y ′′′ − 12 x + 6 e 2 x = 0 22

NOTE: To solve a differential equa- tion having the special form y ( n ) ( x ) = f ( x ) , simply integrate f n times, and EACH integration step produces an arbitrary constant; there will be n independent arbi- trary constants. 23

Intuitively, to find a set of solutions of an n -th order differential equation � � x, y, y ′ , y ′′ , . . . , y ( n ) F = 0 we “integrate” n times, with each in- tegration step producing an arbitrary constant of integration (i.e., a param- eter ). Thus, ”in theory,” an n -th order differential equation has an n -parameter family of solutions . 24

SOL VING A DIFFERENTIAL EQUA- TION: To solve an n -th order differential equa- tion � � x, y, y ′ , y ′′ , . . . , y ( n ) = 0 F means to find an n -parameter family of solutions. ( Note: Same n .) NOTE: An “ n -parameter family of solutions” is more commonly called the GENERAL SOLUTION . 25

Examples: Find the general solution: y ′ − 3 x 2 − 2 x + 4 = 0 1. 26

y = x 3 + x 2 − 4 x + C 27

y ′′ + 2 sin 2 x = 0 2. 28

y = 1 2 sin 2 x + C 1 x + C 2 29

y ′′′ − 3 y ′′ + 3 y ′ − y = 0 3. y = C 1 e x + C 2 xe x + C 3 x 2 e x Answer: x 2 y ′′ − 4 xy ′ + 6 y = 3 x 4 4. y = C 1 x 2 + C 2 x 3 + 3 2 x 4 Answer: 30

PARTICULAR SOLUTION: If specific values are assigned to the arbitrary constants in the general solu- tion of a differential equation, then the resulting solution is called a particular solution of the equation. 31

Examples: y ′′ = 6 x + 8 e 2 x 1. General solution: y = x 3 + 2 e 2 x + C 1 x + C 2 Particular solutions: 32

x 2 y ′′ − 2 xy ′ + 2 y = 4 x 3 2. General solution: y = C 1 x + C 2 x 2 + 2 x 3 Particular solutions: 33

THE DIFFERENTIAL EQUATION OF AN n -PARAMETER FAMILY: Given an n -parameter family of curves. The differential equation of the fam- ily is an n -th order differential equation that has the given family as its general solution. 34

Examples: y 2 = Cx 3 + 4 is the general solu- 1. tion of a DE. a. What is the order of the DE? b. Find the DE? 35

y = C 1 x + C 2 x 3 is the general 2. solution of a DE. a. What is the order of the DE? b. Find the DE? 36

General strategy for finding the dif- ferential equation Step 1. Differentiate the family n times. This produces a system of n +1 equations. Step 2. Choose any n of the equa- tions and solve for the parameters. Step 3. Substitute the “values” for the parameters in the remaining equa- tion. 37

Examples: The given family of functions is the general solution of a differential equa- tion. (a) What is the order of the equation? (b) Find the equation. 38

y = Cx 3 − 2 x 1. (a) (b) 39

y = C 1 e 2 x + C 2 e 3 x 2. (a) (b) 40

3. y = C 1 cos 3 x + C 2 sin 3 x (a) (b) 41

y = C 1 x 4 + C 2 x + C 3 4. (a) (b) 42

y = C 1 + C 2 x + C 3 x 2 5. (a) (b) 43

1.4. INITIAL-VALUE PROBLEMS: 1. Find a solution of y ′ = 3 x 2 + 2 x + 1 which passes through the point ( − 2 , 4); that is, satisfies y ( − 2) = 4. 44

y = x 3 + x 2 + x + C (the general solu- tion) 45

y = x 3 + x 2 + x + 10 (the particular solution that satisfies the 46

2. y = C 1 cos 3 x + C 2 sin 3 x is the general solution of y ′′ + 9 y = 0 . a. Find a solution which satisfies y (0) = 3 47

b. Find a solution which satisfies y (0) = 3 , y ′ (0) = 4 48

y = 3 cos 3 x + 4 3 sin 3 x is the solution of y ′′ + 9 y = 0 , y (0) = 3 , y ′ (0) = 4 . 49

c. Find a solution which satisfies y (0) = 4 , y ( π ) = 4 d. Find a solution which satisfies y (0) = 4 , y ( π ) = − 4 50

An n -th order initial-value problem consists of an n -th order differential equation � � x, y, y ′ , y ′′ , . . . , y ( n ) F = 0 together with n (initial) conditions of the form y ( c ) = k 0 , y ′ ( c ) = k 1 , y ′′ ( c ) = k 2 , . . . , y ( n − 1) ( c ) = k n − 1 where c and k 0 , k 1 , . . . , k n − 1 are given numbers. 51

NOTES: 1. An n -th order differential equation can always be written in the form � � x, y, y ′ , y ′′ , · · · , y ( n ) F = 0 by bringing all the terms to the left- hand side of the equation. 2. The initial conditions determine a particular solution of the differential equation. 52

Strategy for Solving an Initial-Value Problem: Step 1. Find the general solution of the differential equation. Step 2. Use the initial conditions to solve for the arbitrary constants in the general solution. 53

Examples: 1. Find a solution of the initial-value problem y ′ = 4 x + 6 e 2 x , y (0) = 5 54

y = C 1 e − 2 x + C 2 e 4 x is the general 2. solution of y ′′ − 2 y ′ − 8 y = 0 Find a solution that satisfies the initial conditions y (0) = 3 , y ′ (0) = 2 55

y = C 1 x + C 2 x 3 3. is the general solution of y ′′ − 3 x y ′ + 3 x 2 y = 0 a. Find a solution which satisfies y ′ (1) = − 4 . y (1) = 2 , 56

b. Find a solution which satisfies y ′ (0) = 2 . y (0) = 0 , c. Find a solution which satisfies y ′ (0) = 3 . y (0) = 4 , 57

EXISTENCE AND UNIQUENESS: The fundamental questions in a course on differential equations are: 1. Does a given initial-value problem have a solution? That is, do solutions to the problem exist ? 2. If a solution does exist, is it unique ? That is, is there exactly one solution to the problem or is there more than one solution? 58