diff equation 2 2011-fall theory

8/3/2019 Diff Equation 2 2011-Fall Theory

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S.l.dr.ing.mat. Alina BogoiDifferential Equations

CHAPTER 1

Introduction to

Differential Equations

POLITEHNICA University of BucharestFaculty of Aerospace Engineering


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Diff_Eq_2_2011 2

Outline

Definitions and terminology

Initial-value problemsDifferential equations as mathematical

models


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Definition:A differential equation contains the derivates of onedependent variables with respect to one or more

independent variables.

32 += xdx

dy

2

23 0

d y d ya y

d x d x+ + =

6 4z z

x yx y

+ =

Examples:1.

2.

3.

Differential equations

yx

yx

z=

2

2

4.


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General form of nth- order ODE in onedependent variable:

Normal form of

0),,',,()(

=n

yyyxF K

),,',,( )1( = nn

n

yyyxfdx

ydK

General Form ODE


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Example: 1st order ODE The first-order differential equation

contain only y and may contain y and

given function ofx.

( , , ') 0

' ( , )

F x y y

y f x y

=

=

Exemple:xyx =+4

xyxy 4/)( =


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CLASSIFICATION OF DIFFERENTIAL

EQUATIONSDifferential equations are classified

according to

(i)

type

(ii) order(iii) degree

(iv)

linearity


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Basic Concepts

ODE vs. PDE Dependent Variables vs.

Independent Variables

( i )Type


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CLASSIFICATION BY TYPE

Differential equations are divided into two types.1. An equation involving only derivatives of a

single independent variable is called anordinary differential equation (ODE).2. An equation involving the partial derivativesof one or more dependent variables oftwo

or more

independent variables is called a

partial differential equation (PDE).


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Basic Concepts PartialDifferential Equations

An unknown function (dependent variable)of two or more independent variables (e.g.

x and y)

6 4z z

x y

x y

+ =

2 2

2 2 0

u u

x y

+ =


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Basic Concepts

The order of the differential

equation is order of the highest

derivative in the differential equation.

(i i ) Order


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Order of Differential Equation

Differential Equation ORDER

32 += xdxdy

0932

2

=++ ydxdy

dxyd

36

4

3

3

=+

+ y

dx

dy

dx

yd

1

2

3


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Basic Concepts

Thedegreeof a di f f erent ial

equat ion is t hepower of t hehighest order der ivat ive t ermin

t he di f f erent ial equat ion.

(i i i ) Degree


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Degree of Differential Equation

Differential Equation Degree

032

2

=++ aydx

dy

dx

yd

36

4

3

3

=+

+ y

dxdy

dxyd

03

53

2

2

=+

+

dx

dy

dx

yd

1

1

3


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In general, an nth-order differential equation is

said to be linear

if it can be written in the form

).()()()()( 011

1

1 xgyxadx

dyxa

dx

ydxa

dx

ydxa

n

n

nn

n

n =++++

L

Two properties of a linear ODE:

1) y, y

, y(n)

are of the first degree.2) Coefficients a0, a1, , an depend at moston x.

(iv) L inear i t y


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Linear Differential Equation

36

4

3

3

=+

+ y

dxdy

dxyd

is non - linear

becausethe 2nd term is

not of degreeone.

2

23 9 0.

d y dyx y

dx dx

+ + =

Examples:

is linear.1

1

33

2

22 x

dx

dyy

dx

ydx =+

is non - linearbecauseof the 2nd term


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The following cases are for n = 1,2 and

n = 1,4

)()()(01

xgyxadx

dyxa =+

)()()()(012

2

2xgyxa

dx

dyxa

dx

ydxa =++

xeyyy =+ 2')1(

024

4

=+ydx

yd

Solution of an ODE


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Any function , defined on an intervalIand possessing at

least n

derivatives that are continuous onI, when

substituted into an nth-order ODE reduces the equationto an identity.

DEFINITION Solution of an ODE

Solution of an ODE

0),,',,( )( =nyyyxF K

SOLUTION OF A DIFFERENTIAL


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SOLUTION OF A DIFFERENTIAL

EQUATION

NOTE: Depending on the context of the problem

the intervalI

could be

an open interval,

a half-open interval,

or an infinite interval.

A solution of ODE is a function thatpossesses at least n derivatives :

F(x, (x), (x), , (n)(x)) = 0 for all x

I.


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A solution of a given first-order differential

equation (*) on some open interval a


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PARTICULAR SOLUTIONS

DEF. 1.A solution of a differential equation that is free

of arbitrary parameters is called aparticularsolution.

DEF. 2.

A differential equation may have an additional

solution that cannot be obtained from thegeneralsolution and is then called aparticular solution.


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Basic Concept General solution vs. Particular

solution

General solution

arbitrary constant c Particular solution choose a specific c

,....2,3

'

=

+=

=

c

csinxy

cosxy


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Basic Concept Particular solutions

Example

The general solution : y=cx-c2

A particular solution : y=x2/4

0' =+ yxyy'2


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Initial-value Problems

Introduction: A solution y(x) of a DE satisfies aninitial condition.

Example: On some interval I containing xo,solve

subject to

(1)

This is called an Ini t ial-Value Problem(IVP). y(xo) = yo , y(xo) = y1 ,,

are called ini t ial condi t ions.

),,',,( )1( = nn

n

yyyxfdx

ydK

10)1(

1000 )(,,)(',)( === n

n yxyyxyyxy K

10)1( )(

= nn yxy


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Basic Concept

Def: A differential equation

together with an initial conditionis called an initial value

problem(or a Cauchy problem)

0 0

' ( , )

( )

y f x y

y x y

=

=


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First and Second Order IVPs

and

00 )(:

),(:

yxytosubject

yxfdx

dysolve

=

=

2

2

0 0

0 1

: ( , , ')

: ( ) ,

'( )

d ysolve f x y y

dx

subject to y x y

y x y

=

=

=


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Theorem (Existence and uniqueness):

The I.V.P. always has a unique solution in

a rectangle containing the point (x0, y0), iff and fx are continuous there.

Basic Concept

First Order Ordinary Differential


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equation

Some Mathematical Models


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Some Mathematical Models

TERMINOLOGYAmodelstarts by

(i) identifying the variables that are responsiblefor changing the system(ii) a set of reasonable assumptions about thesystem.

The mathematical construct of all theseassumptions is called amathematical model

and is often a differential equation or system

of differential equations.

MODEL OF A FREELY


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O O

FALLING BODY

.)0(,)0( 00

2

2

ssvv

gdt

sd

==

=

VIBRATION OF A MASS ON A SPRING


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VIBRATION OF A MASS ON A SPRING

Hookes Law: The restoring force of a stretched

spring is opposite to the direction of elongation

and is proportional to the amount of elongation.

That is, k(s

+x)

2

2

)(dt

xdmxskmg =+

VIBRATION OF A MASS


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kxksmgkxmg)xs(k

dt

xdm =+=++=

43421zero

2

2

V ON O M SSON A SPRING


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Ser ies Circui t s


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From Kirchhoffs second law, we have

where dq(t)/dt = i(t), which is the current.

)(1

2

2

tEqCdt

dqRdt

qdL =++

q (t) is charge on capacitor,L is inductance,

C is capacitance.

R is resistance andE is voltage


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Series Circuits

POLITEHNICA University of Bucharest


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S.l.dr.ing.mat. Alina

Bogoi

Differential

Equations

Faculty of Aerospace Engineering

CHAPTER 2

First-Order Differential

Equations


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Outline Separation of Variables

Exact equations

Integrating Factors

Homogenous equations


equation

Separation of variables


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A first-order DE of the form dy/dx = g(x)h(y) is said to be separable or to have separable variables.

DEFINITION Separable Equations

Rewrite )()( xgdxdyyp =

c)x(G)y(Hordx)x(gdy)y(p +==



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A separable differential equation can be expressed asthe product of a function ofx and a function ofy.

( ) ( )dy g x h ydx = Solve:

2

2dy

xydx =

( ) 0h y

(1 ) - 0.x dy y dx+ =

( )22

2 1xdy

x y edx = + ( )2

tanx

y e C = +

= +6 exy x

Differential of a Function of Two


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Differential of a Function of Two

Variables Ifz = f(x, y), its differentialor total

differentialis

Now ifz = f(x, y) = c,

eg: ifx2 5xy + y3 = c, then(2x 5y) dx + (-5x + 3y2) dy = 0

dyyfdx

xfdz

+

=

0=

+

dy

y

fdx

x

f

Exact Equations


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An expression M(x,

y)

dx

+ N(x,

y)

dy

is an exact

differentialin a regionR corresponding to thedifferentialof some functionf(x,

y). A first-order DE

of the formM(x, y) dx + N(x, y) dy = 0 is said to be an exact equation, if the left side is an

exact differential.

DEFINITIONExact Equation

q

Exact Equations


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Consider a first-order DE of the formM(x,

y)

dx

+ N(x,

y)

dy

= 0

Let M(x, y) andN(x,y)be continuous and havecontinuous first partial derivatives in a regionR definedby a

diff equation 2 2011-fall theory

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