Math 3120 Differential Equations with Boundary Value Problems Chapter 1 Introduction to Differential Equations.

Math 3120 Differential Equations

withBoundary Value

Problems

Chapter 1Introduction to Differential

Equations

Basic Mathematical ModelsMany physical systems describing the real world

are statements or relations involving rate of change. In mathematical terms, statements are equations and rates are derivatives.Definition: An equation containing derivatives is called a differential equation. Differential equation (DE) play a prominent role in physics, engineering, chemistry, biology and other disciplines. For example: Motion of fluids, Flow of current in electrical circuits, Dissipation of heat in solid objects, Seismic waves, Population dynamics etc.Definition: A differential equation that describes a physical process is often called a mathematical model.

Formulate a mathematical model describing motion of an object falling in the atmosphere near sea level.

Variables: time t, velocity v Newton’s 2nd Law: F = ma = net

force

Force of gravity: F = mg downward force

Force of air resistance: F = v upward force

Then

vmgdt

dvm

Basic Mathematical Models

dt

dvm

We can also write Newton’s 2nd Law:

where s(t) is the distance the body falls in time t from its initial point of release

Then,

Basic Mathematical Models

dt

dvs

dt

dsmF re whe

2

2

mgdt

ds

dt

sdm

2

2

(1)

(2)

(3)

(4)

(5)

mgdt

ds

dt

sdm

2

2

Examples of DE

vmgdt

dvm

equation) (wave ),(),(

equation)(heat ),(),(

2

2

2

22

2

2

22

t

txu

x

txua

t

txu

x

txu

)(1

2

2

tEqCdt

dsR

dt

qdL

Classifications of Differential Equation

By Types Ordinary Differential Equation (ODE) Partial Differential Equation (PDE)

Order Systems Linearity

Linear Non-Linear

Ordinary Differential Equations

When the unknown function depends on a single independent variable, only ordinary derivatives appear in the equation. In this case the equation is said to be an ordinary differential equations.

For example:

A DE can contain more than one dependent variable. For example:

05.0,2.08.92

2

ydx

dy

dx

ydv

dt

dv

yxdt

dy

dt

dx

Partial Differential Equations

When the unknown function depends on several independent variables, partial derivatives appear in the equation. In this case the equation is said to be a partial differential equation.

Examples:

equation) (wave ),(),(

equation)(heat ),(),(

2

2

2

22

2

2

22

t

txu

x

txua

t

txu

x

txu

Notation

Leibniz

Prime

Dot

Subscript

)()1()4( ,,,,, nn yyyyyy

n

n

dx

yd

dx

yd

dx

yd

dx

dy,........,,

3

3

2

2

ydx

ydy

dx

dy 2

2

,

yyxxx uuu ,,

Systems of Differential Equations

Another classification of differential equations depends on the number of unknown functions that are involved.

If there is a single unknown function to be found, then one equation is sufficient. If there are two or more unknown functions, then a system of equations is required.

For example, Lotka-Volterra (predator-prey) equations have the form

where u(t) and v(t) are the respective populations of prey and predator species. The constants a, c, , depend on the particular species being studied.

uvcvdtdv

uvuadtdu

/

/

Order of Differential Equations

The order of a differential equation is the order of the highest derivative that appears in the equation.

Examples:

An nth order differential equation can be written as

The normal form of Eq. (6) is

tuuedt

yd

dt

ydyy yyxx

t sin 1 03 22

2

4

4

)7( ,,,,,,)( )1()( nn yyyyytfty

(6) 0,,,,,, )( nyyyyytF

Linear & Nonlinear Differential Equations

An ordinary differential equation

is linear if F is linear in the variables

Thus the general linear ODE has the form

The characteristic of linear ODE is given as

0,,,,,, )( nyyyyytF

)(,,,,, nyyyyy

)()()()( )1(1

)(0 tgytaytayta n

nn

Linear & Nonlinear Differential Equations

Example: Determine whether the equations below are linear or nonlinear.

tuuutyy

tuuutyey

tdt

ydt

dt

ydyy

yyxx

yyxxy

cos)sin()6(023)3(

sin)5(023)2(

1)4(03)1(

2

22

2

4

4

Solutions to Differential Equations

A solution of an ordinary differential equation

on an interval I is a function (t) such that

exists and satisfies the equation:

for every t in I.

Unless stated we shall assume that function f of Eq. (7) is a real valued function and we are interested in obtaining real valued solutions

NOTE: Solutions of ODE are always defined on an interval.

)1()( ,,,,,)( nn tft

)()1( ,,,, nn )7( ,,,,,,)( )1()( nn yyyyytfty

)(ty

Solutions to Differential Equations

Example: Show that is a solution of the ODE on the interval (-∞, ∞).

Verify that is a solutions of the ODE on the interval (-∞, ∞).

tty sin)(

tty cos)( 0 yy

0 yy

Types of Solutions

Trivial solution: is a solution of a differential equation that is identically zero on an interval I.

Explicit solution: is a solution in which the dependent variable is expressed solely in terms of the independent variable and constants. For example,

are two explicit solutions of the ODE

Implicit solution is a solution that is not in explicit form.

ttytty sin)( and ,cos)( 0 yy

Families of Solutions

A solution of a first- order differential equation

usually contains a single arbitrary constant or parameter c.

One-parameter family of solution: is a

solution containing an arbitrary constant represented by a set of solutions.

Particular solution: is a solution of a differential equation that is free of arbitrary parameters.

0,, yyxF

0,, cyxG

Initial Value Problems (IVP)

Initial Conditions (IC) are values of the solution and /or its derivatives at specific points on the given interval I.

A differential equation along with an appropriate number of IC is called an initial value problem. Generally, a first order differential equation is of the type

An nth order IVP is of the form

where are arbitrary constants. Note: The number of IC’s depend on the order of the DE.

10)1(

1000

)1()(

)(,....,)(',)( subject to

),.....,',,(

nn

nn

ytyytyyty

yyytfy

00 )( ),,(' ytyytfy

110 ,....,, nyyy

Solutions to Differential Equations

Three important questions in the study of differential equations: Is there a solution? (Existence)

If there is a solution, is it unique? (Uniqueness)

If there is a solution, how do we find it?

(Qualitative Solution, Analytical Solution, Numerical Approximation)

Theorem 1.2.1: Existence of a Unique Solution

Suppose f and f/y are continuous on some open rectangle R defined by (t, y) (, ) x (, ) containing the point (t0, y0). Then in some interval (t0 - h, t0 + h) (, ) there exists a unique solution y = (t) that satisfies the IVP

It turns out that conditions stated in Theorem 1.2.1 are sufficient but not necessary.

00 )( subject to

),('

yty

ytfy