Unit i Introduction to Differential Equations

7/24/2019 Unit i Introduction to Differential Equations

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INTRODUCTION TO

DIFFERENTI L EQU TIONS

BCE1-240


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Agenda

I. IntroductionII. Definition and terminology

III. Initial value problems

IV. Differential equations as mathematical models

INTRODUCTION TO DIFFERENTIAL EQUATIONS1/11/2016


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Introduction

Leibniz notation:

,

,

, .

=

=

Lagrange or Prime notation:

, , , but is () and the general form is ()



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Introduction

Example: Given an equation written in Leibniz notation, rewusing prime notation.

I.

+ 6 = 0

II.

+

= 2 +

Leibniz notation is preferred. It clearly displays both the dependeindependent variable



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Introduction

In this course one of the task will be to solve differential equsuch as + 2 + = 0 for an unknown function = (

The derivative / of a function = () is itself anothefunction () found by an appropriate rule.

If given = .then

= 0.2 by using derivative rules

What about if given

= 0.2 to find ?



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Definition and terminology

Differential equation (DE): An equation containing the derivatives of one or more dependent

with respect to one or more independent variables, is said to be aequation.

DE are classified by:

Type

Order

Linearity



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CLASSIFICATION BY TYPE:I. Ordinary differential equation (ODE): contains only ordinary de

one or more dependent variables with respect to a single indepvariable

II. Partial differential equation (PDE). involves partial derivatives omore dependent variables of two or more independent variable



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CLASSIFICATION BY ORDER The order of a DE is the order of the highest derivative in the equa

What about these?



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CLASSIFICATION BY LINEARITY

I. Linear

An nth-order ordinary differential equation is said to be linear if is line(,, , ). This means that an nth-order ODE is linear when (4) is:

() + () + + + = 0

Properties: The dependent variable and all its derivative , , , () are of the first degre

each term involving is 1). The coefficients , , , of y,

, , depend at most on the independent

INTRODUCTION TO DIFFERENTIAL EQUATIONS

General form of a Differ

, , , , = 0

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I. Linear

Example:



, , , , = 0

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II. Nonlinear

A nonlinear ordinary differential equation is simply one that is notNonlinear functions of the dependent variable or its derivatives, s

or

, cannot appear in a linear equation.



, , , , = 0

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SOLUTION OF AN ORDINARY DE

We assume that it is possible to solve an ordinary differential equation i(4) uniquely for the highest derivative in terms of the remaining variables. That is:

= , , , , normal form

Example:I. 4 + = II. + 6 = 0



, , , , = 0

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SOLUTION OF AN ORDINARY DE

Definition: Any function , defined on an interval I and possessing at leastderivatives that are continuous on I, which when substituted into an nthordinary differential equation reduces the equation to an identity, is saidsolution of the equation on the interval.

In other words, a solution of an nth-order ordinary differential equationfunction that possesses at least n derivatives and for which:

, , , , () = 0 for all in I

We say that satisfies the differential equation on I



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EXAMPLE: Verify that the indicated function is a solution of differential equation on the interval ,

a)

= /; =

b) 2 + = 0;



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EXPLICIT AND IMPLICIT SOLUTIONS

Explicit: A solution in which the dependent variable is expressed sterms of the independent variable and constants

=

, = , =

= /, 2 + = 0, + = 0

Implicit: there exists at least onefunction that satisfies the relatiothe differential equation on I

+ = 25 is an implicit solution of the DE

=



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Initial value problems

Find a solution to satisfy certain prescribed side conditions. conditions imposed on the unknown function () and its dat a point

S:

= , , , ,

S : = , () = , , () =

Where:

, , , are arbitrary real constants and the ,

, , () are called initial conditi



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EXAMPLE:

Solve:

= ,

Subject to: =



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EXAMPLE:

Solve:

= , ,

Subject to: = , =



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EXAMPLE 1: given =

as a solution for the ODE

= 0 = 3 1 = 2




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It is often desirable to describe the behavior of some real-lifor phenomenon, whether physical, sociological, or even ecomathematical terms.

Construction of a mathematical model of a system starts wi

a. Identification of the variables that are responsible for changing t

b. Make a set of reasonable assumptions, or hypotheses, about theare trying to describe.

D.E. AS MATHEMATICAL MODELS



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D.E. AS MATHEMATICAL MODELS



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Analytical Solution to the Falling Parachutist Problem

Falling bodies and air resistance

1/11/2016 INTRODUCTION TO DIFFERENTIAL EQUATIONS

Problem Statement: A parachutist of mass 68.1 kg jum

of a stationary hot air balloon. Use Eq. (1.10) to compu

velocity prior to opening the chute.

The drag coefficient is equal to 12.5 kg/s.

K = proportionality constant called the drag coefficient (kg/s)

=

F= +

=

=

=

( 1

)


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#1

EXERCISES 1.1: 1,3,5,7,9,10,11,13,15,17

#2

EXERCISES 1.1: 23

EXERCISES 1.2: 1,2,3,5,7,9,11,13,35-38

Homework


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Unit i Introduction to Differential Equations