01 Introduction - Copy

7/30/2019 01 Introduction - Copy

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Differential Equations

MATH C241

Text Book: Differential Equations

with Applications and

Historical Notes:

Class hours: T Th S 2

(9.00 A.M. to 9.50 A.M.)

by George F. Simmons

(Tata McGraw-Hill) (2003)


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In this introductory lecture, we

Define a differential equation Explain why we study a differential equation

Define the order and degree of a DE.

Define the solution of a DE

Formation of a DE

Discuss the Orthogonal trajectories of a family of

curves.


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Many important and significant

problems in engineering, the

physical sciences, and the social

sciences, when formulated in

mathematical terms require thedetermination of a function satisfying

an equation containing derivatives of

the unknown functions. Suchequations are called differential

equations i.e.


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A dif ferent ial equation is a

relationship between an independentvariable, (let us say x), a dependent

variable (let us call this y), and one or

more derivatives of y with respect tox. 2

3

2 0xd y dy

x y edxdx

is a differential equation.


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we recall that y= f(x) is a given function,

then its derivatives dy/dx can be

interpreted as the rate of change of ywith respect to x. In any natural

process, the variables involved and their

rates of change are connected with oneanother by means of the basic scientific

principles that govern the process.When this connection is expressed in

mathematical symbols, the result is

often a differential equation.


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Perhaps the most familiar example

is Newtons law2

2

d xm F

dt

For the position x(t) of a particle acted

on by a force F. In general F will be afunction of time t, the position x, and the

velocity dx/dt.


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To determine the motion of a particle

acted on by a given force F it is

necessary to find a function x(t) satisfythe above equations. If the force is that

due to gravity, the F = - mg and

2

2

d xm mg

dt


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For example, the distances traveled in time t

by a freely falling body of mass m satisfies

the DE2

2

d sg

dt

The time rate of change of a population P(t) with

constant birth and death rates is, in many simple cases,

proportional to the size of the population. That isdP

kPdt

Where k is the constant of proportionality


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The other examples of Physical

phenomena involving rates of changeare :

Motion of fluids Motion of mechanical systems

Flow of current in electrical circuitsA DE that describes a physical process is

often called a Mathematical Model.


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Ordinary Differential Equations:

An ordinary differential equation (ODE) isa differential equation that involves the

(ordinary) derivatives or differentials of

only a single independent variable.

equations are ODEs, while

is not ODE.

y'' - 2y' + y = cos

sinx

x

dye xdx

2 2

2 2

u u u

tx y


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In fact, the above equation is a

partial differential equation. A partialdifferential equation (PDF) is adifferential equation that involves the

partial derivatives of two or moreindependent variables.

22

2w wa

tx

Heat equation


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Order

The order of a differential equation isjust the order of highest derivativeused.

02

2

dt

dy

dt

yd

.2nd order

3

3

dt

xdx

dt

dx 3rd order


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Degree of a Differential Equation

The power of the highest order

derivative occurring in a differential

equation, after it is free fromradicals and fractions, is called the

degree of a differential equation.


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2 3 2 2 2 2{1 ( ) } ( / )dy

a d y dxdx

Example: The equation

22 3 / 2

2{1 ( ) }

dy d ya

dx dx

is of second order and the second degree

as the equation can be written as


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More generally, the equation

( )( , , , , ) 0nF x y y y is an ODE of the nth order. Equation (1)

represents a relation between the n+2

variables x, y, y, y, ., y(n) which under

suitable conditions can be solved for y

(n)

in terms of the other variables:

(1)

( ) ( 1)

( )

n n

y f x y y y

(2)


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Initial-Value Problem:

A differential equation along with

subsidiary conditions on the unknown

function and its derivatives, all givenat the same value of the independent

variable, constitutes an initial-value

problem and the conditions are initial

conditions.


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For example: The problem

y'' + 2y' = ex; y(p) = 1, y'(p) = 2

is an initial-value problem,because

the two subsidiary conditions areboth given at x = p


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Boundary-Value problem:

If the subsidiary conditions are given

at more than one value of the

independent variable, the problem isa boundary-value problem and the

conditions are boundary conditions.


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For Example:The problem

y'' + 2y' = ex; y(0) =1, y'(1) = 1

is a boundary-value problem, because

the two subsidiary conditions are

given at the different values x = 0 andx = 1.


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A DE is said to be linear when the

dependent variable and all the derivativesof it appear only in the 1st degree.

Examples

x

dx

dy2.1

ydx

yd

2

2

.2

linear

linear


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xeydxdy

dxyd 32

2

65.3

Examples (Continued)

2 24. 1 0y y y

linear

Non linear


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Solutions of ODEs:

A solution of an ODE( )( , , , , ) 0 (1)nF x y y y

on the interval [a, b] is a function f such that

f, f, f, .f(n) exist for all x[a, b] and

( ), ( ), ( ), ( ) 0nF x f x f x f x

for all x[a, b].


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Given a DE any relation between the

variables (that is free from derivatives) that

satisfies the DE is called the solution of the

DE

For example is a solution of the DE2y x

2dy xdx


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2 2 4x y is a solution of the DE

0xdx y dy sin2x t is a solution of the DE

2

24

d xx

dt

(Note here tis the independent variable and

x ia function oft.)


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If no initial conditions are given, we call

the description of all solutions to the

differential equation the general solution.

General and particular solution:

cos siny' x y x c general solution

sin 2 siny x or y x particaular solution


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It is clear that the general solution of the

DE 2dy xdx

is the one-parameter family of parabolas2y x c

c isan arbitrary constant.

(See the figure in the next slide.)


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Figure 1 Graphs of 2y x C for various

value of C


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It can be shown that the general solution

of the DE

is the two-parameter family of curves

1 2cos 2 sin 2x c t c t

where c1, c2are arbitrary constants.

2

24

d xx

dt


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Conversely, given a family of curves, we can

find the DE satisfied by the family (by

eliminating the parameters by differentiation).

Consider the one-parameter family of curves

2y c xDifferentiating w.r.t. x, we get 2

dyc x

dx

Eliminating c, we get the DE of the family as

2dy

x y

dx


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Find the DE of the family of all circles

tangent to they-axis at the origin

Ca

Solution The equation

to the circle tangent to

y-axis at the origin isgiven by

2 2 2( )x a y a

or2 22 0x ax y


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or 0x a yy

Eliminating a, we get the DE of the family as2 2

02

x yx yy

x

or

2 2

2

y xy

xy

Differentiating w.r.t. x, we get

2 2 2 0x a yy

i.e.

2 2

0

2

x yyy

x


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Consider the two-parameter family of curves2 2

2 2 1

x y

a b Differentiating w.r.t. x, we get

2 2

2 20

x y dy

a b dx

Eliminating a,b, we get the DE of the family as

Again differentiating w.r.t. x, we get22

2 2 2

2 20

d y dyy

a b dx dx

22

20

d y dy dyxy x y

dx dx dx


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Example: Consider the DE xdy

xedx

The general solution isx xy xe e c

where c is an arbitrary constant.

We now show that there is a unique solutionsuch that whenx = 1,y =3.

Replacing x by 1, y by 3, we get a

unique c, namely, c = 3.Thus the desired unique solution is

3

x x

y xe e


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We now state (without proof) a theorem

which asserts that under suitable conditions

that a first order DE

( , )dy

f x y

dx

has a unique solutiony =g(x) satisfying the

initial conditions: whenx = x0,y =y0


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Existence and uniqueness of solution of a

first order initial-value problem

Picards Theorem

Consider the first order d.e. ( , )dy

f x ydx

Suppose ( , )f x y andf

y

are both

continuous (as functions ofx, y) at each

point (x,y) on and inside a closed rectangle

R of the x-y plane. Then for each point


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(x0,y0) inside the rectangle R, there exists a

unique solution y = g(x) of the above DE

such that whenx =x0,y =y0.

Geometrically speaking, through each point

(x0,y0) inside the rectangle R, there passes aunique solution curvey =g(x)of the DE

( , )dy

f x ydx


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(x0,y0)

y=g(x)

R


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Orthogonal trajectories of a family of

curves

Consider two families of curves, , in the

xy plane. Suppose every curve in the family

intersects every curve in the family

orthogonally (i.e. the angle between the two

curves at each point of intersection is 90o, i.e.

a right angle), then each family is said to be afamily of orthogonal trajectories of the other

family.


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For example if is the family of all cirles

centre at the origin and is the family of

all lines through the origin, then we easilysee that each is the family of orthogonal

trajectories of the other.


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If the DE of a one-parameter family of

curves in the xy plane is given by

( , )dy f x ydx

from definition, it trivially follows that the DEof the family of orthogonal trajectories is

given by1

( , )

dy

dx f x y

Integrating the above DE we get the algebraic

equation of the family of orthogonal trajectories.


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Example Consider the one parameter family

of parabolas having the focus at the origin:

2 4 ( )y c x c

The DE of the above family is:

2 2

y yyx

y

(*)

c > 0 c < 0


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Hence the DE of the family of orthogonal

trajectories is got by replacing

And hence is given by

2 2

yy yx

y

1y by

y

2 2

y yyx

y

or

which is same as (*).

Hence the family of orthogonal trajectories isthe given family of parabolas itself. Or we say

that the given family of parabolas is self-

orthogonal.


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In the next lecture we discuss the

methods of solving first orderdifferential equations.

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