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7/24/2019 Unit i Introduction to Differential Equations
1/23
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
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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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Definition and terminology
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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Definition and terminology
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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Definition and terminology
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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Definition and terminology
CLASSIFICATION BY LINEARITY
I. Linear
Example:
INTRODUCTION TO DIFFERENTIAL EQUATIONS
General form of a Differ
, , , , = 0
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Definition and terminology
CLASSIFICATION BY LINEARITY
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.
INTRODUCTION TO DIFFERENTIAL EQUATIONS
General form of a Differ
, , , , = 0
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Definition and terminology
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
INTRODUCTION TO DIFFERENTIAL EQUATIONS
General form of a Differ
, , , , = 0
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Definition and terminology
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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Definition and terminology
EXAMPLE: Verify that the indicated function is a solution of differential equation on the interval ,
a)
= /; =
b) 2 + = 0;
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Definition and terminology
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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Initial value problems
EXAMPLE:
Solve:
= ,
Subject to: =
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Initial value problems
EXAMPLE:
Solve:
= , ,
Subject to: = , =
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EXAMPLE 1: given =
as a solution for the ODE
= 0 = 3 1 = 2
Initial value problems
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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
INTRODUCTION TO DIFFERENTIAL EQUATIONS1/11/2016
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