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

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

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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

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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

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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.

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  • Integration:
  • f(x)dx,

b

a f(x)dx, F(x) :=

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

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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.

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Examples: 1. y′ = 2x + cos x 2. dy dt = ky (exponential growth/decay) 3. x2y′′ − 2xy′ + 2y = 4x3

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4. ∂2u ∂x2 + ∂2u ∂y2 = 0 (Laplace’s eqn.) 5. d3y dx3 − 4d2y dx2 + 4dy dx = 0

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TYPE: If the unknown function depends on a single independent variable, then the equation is an

  • rdinary differential equation (ODE).

If the unknown function depends on more than one independent variable, then the equation is a partial differential equation (PDE).

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

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Examples: 1. y′ = 2x + cos x 2. dy dt = ky (exponential growth/decay) 3. x2y′′ − 2xy′ + 2y = 4x3

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4. ∂2u ∂x2 + ∂2u ∂y2 = 0 (Laplace’s eqn.) 5. d3y dx3 − 4d2y dx2 + 4dy dx = 0 6. d2y dx2+2x sin

dy

dx

  • +3exy = d3

dx3(e2x)

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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.

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Examples: 1. y′ = 2x + cos x

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2. y′ = ky

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3. y′′ − 2y′ − 8y = 4e2x Is y = 2e4x − 1

2 e2x

a solution?

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y′′ − 2y′ − 8y = 4e2x Is y = e−2x + 2e3x a solution?

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4. x2y′′ − 4xy′ + 6y = 3x4 Is y = 3

2 x4 + 2x3

a solution?

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x2y′′ − 4xy′ + 6y = 3x4 Is y = 2x2 + x3 a solution?

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5. ∂2u ∂x2 + ∂2u ∂y2 = 0 u = ln

  • x2 + y2

Solution?

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∂2u ∂x2 + ∂2u ∂y2 = 0 u = cos x sinh y, u = 3x − 4y Solutions??

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

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7. Find a value of r, if possible, such that y = xr is a solution of x2y′′ + 2x y′ − 6y = 0.

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8. Find a value of r, if possible, such that y = xr is a solution of y′′ − 1 x y′ − 3 x2 y = 0.

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

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1.3. n-PARAMETER FAMILY OF SOLUTIONS / GENERAL SOLU- TION Example: Solve the differential equa- tion: y′′′ − 12x + 6e2x = 0

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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.

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Intuitively, to find a set of solutions of an n-th order differential equation F

  • x, y, y′, y′′, . . . , y(n)
  • = 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.

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SOL VING A DIFFERENTIAL EQUA- TION: To solve an n-th order differential equa- tion F

  • x, y, y′, y′′, . . . , y(n)
  • = 0

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.

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Examples: Find the general solution: 1. y′ − 3x2 − 2x + 4 = 0

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y = x3 + x2 − 4x + C

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2. y′′ + 2 sin 2x = 0

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y = 1 2 sin 2x + C1x + C2

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3. y′′′ − 3y′′ + 3y′ − y = 0 Answer: y = C1ex + C2xex + C3x2ex 4. x2y′′ − 4xy′ + 6y = 3x4 Answer: y = C1x2 + C2x3 + 3

2 x4

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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.

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Examples: 1. y′′ = 6x + 8e2x General solution: y = x3 + 2e2x + C1x + C2 Particular solutions:

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2. x2y′′ − 2xy′ + 2y = 4x3 General solution: y = C1x + C2x2 + 2x3 Particular solutions:

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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.

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Examples: 1. y2 = Cx3 + 4 is the general solu- tion of a DE.

  • a. What is the order of the DE?
  • b. Find the DE?

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2. y = C1x + C2x3 is the general solution of a DE.

  • a. What is the order of the DE?
  • b. Find the DE?

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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.

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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.

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1. y = Cx3 − 2x (a) (b)

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2. y = C1e2x + C2e3x (a) (b)

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3. y = C1 cos 3x + C2 sin 3x (a) (b)

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4. y = C1x4 + C2x + C3 (a) (b)

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5. y = C1 + C2x + C3x2 (a) (b)

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1.4. INITIAL-VALUE PROBLEMS: 1. Find a solution of y′ = 3x2 + 2x + 1 which passes through the point (−2, 4); that is, satisfies y(−2) = 4.

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y = x3 + x2 + x + C (the general solu- tion)

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y = x3 + x2 + x + 10 (the particular solution that satisfies the

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2. y = C1 cos 3x + C2 sin 3x is the general solution of y′′ + 9y = 0.

  • a. Find a solution which satisfies

y(0) = 3

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

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y = 3 cos 3x + 4

3 sin 3x

is the solution

  • f

y′′ + 9y = 0, y(0) = 3, y′(0) = 4.

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

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An n-th order initial-value problem consists of an n-th order differential equation F

  • x, y, y′, y′′, . . . , y(n)
  • = 0

together with n (initial) conditions of the form y(c) = k0, y′(c) = k1, y′′(c) = k2, . . . , y(n−1)(c) = kn−1 where c and k0, k1, . . . , kn−1 are given numbers.

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NOTES: 1. An n-th order differential equation can always be written in the form F

  • x, y, y′, y′′, · · · , y(n)
  • = 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.

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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.

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Examples: 1. Find a solution of the initial-value problem y′ = 4x + 6e2x, y(0) = 5

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2. y = C1e−2x + C2e4x is the general solution of y′′ − 2y′ − 8y = 0 Find a solution that satisfies the initial conditions y(0) = 3, y′(0) = 2

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3. y = C1x + C2x3 is the general solution of y′′ − 3 x y′ + 3 x2 y = 0 a. Find a solution which satisfies y(1) = 2, y′(1) = −4.

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

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EXISTENCE AND UNIQUENESS: The fundamental questions in a course

  • n 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?

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