differential.equations

8/10/2019 Differential.equations

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Linear Differential

Equations - Intro

4thWeek

Semester 3 2011/12 Session


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Basic Concepts

Definition 1.1.1:An equation containingthe derivatives of one or more dependentvariables with respect to one or moreindependent variables, is said to be adifferential equation (DE).

Classifications

Type: Ordinary DE or Partial DE

Order

Linearity


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Classifications - Type

Ordinary differential equation (ODE)If an

equation contains only ordinary derivatives of one ormore dependent variables with respect to a singleindependent variable.

Partial differential equation (PDE)An equationinvolving partial derivatives of one or more dependentvariables of two or more independent variables.


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Classifications - Order

The order of a differential equation(either ODE or PDE) is the order of thehighest derivative in the equation.


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Classifications - Linearity

An n-thorder ODE of the form

is said to be linear if the dependent variable y and all its derivatives y,y,y..y(n)

are of the first degree, i.e. the power of each term involving y

is 1. the coefficients a0, a1, a2, anof y,y,y..y(n)depend at most

on the independent variable x.

The degree of an ODE is the power to which the highestorder derivative of a linear ODE is raised.


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Other important notes


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Homogeneous


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Solutions

Definition 1.1.2: Solution of an ODEAny function , defined on an interval Iand

possessing at least nderivatives that arecontinuous on I, which when substituted into

an nth order ODE reduces the equation to anidentity, is said to be a solution(sometimesreferred to as an integralof the equation) ofthe equation on the interval.


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Families of Solutions

Families of Solutions: Solving a DE is essentially evaluating

an integral or antiderivative. For an indefinite integral, anarbitrary integral constant is obtained. Analogously, whensolving a first-order DE , the solution usuallycontains a single arbitrary constant or parameter ci.e.

A set of solutions is called one-parameterfamily of solutions. Similarly when solving an nth-order DE,

, the solution is n-parameter family ofsolutions . Thus a single DE canpossess an infinite number of solutions corresponding to theunlimited number of choices for the parameter(s).

A solution of a DE that is free of arbitrary parameters is calleda particular solutionor particular integral. A singularsolutionis a solution that is not a member of a family ofsolutions i.e. solution that cannot be obtained by specializingany of the parameters in the family of solutions.


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Families of Solutions

9 ' 4 0yy x

1 19 ' 4 9 ( ) '( ) 4yy x dx C y x y x dx xdx C

2 2

1This yields where .4 9 18

Cy xC C

Example

Solution

The solution is a family of ellipses.

22 2 2 2

1 1 1

9

9 2 2 9 4 22

y

ydy x C x C y x C

Observe that given any point (x0,y0),

there is a unique solution curve of theabove equation which curve goes

through the given point.


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IVP and BVP

Initial-value problem (IVP)is a set of conditions

specified at the same value of the independentvariable, which are imposed on the dependentvariable and its derivatives. For an nth order linear DE

which is subject to ,the solution is a function defined on some interval Icontaining and satisfies the initial conditions given.

If the set of conditions imposed on the dependent

variable and its derivative are specified at differentpoints , then it iscalled boundary-value problem (BVP).


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Superposition of solutions


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Dependency


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Wronskian


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General Solution


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Superposition of Solutions


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Physical Origin

1. Free falling stone g

dt

sd

2

2

where s is distance or height and g is acceleration due to gravity.

2. Spring vertical displacement ky

dt

ydm

2

2

where y is displacement, m is mass and k is spring constant

3. RLCcircuit, Kirchoff s Second Law

Eqcdt

dqR

dt

qdL

12

2 q is charge on capacitor, L isinductance, c is capacitance,

R is resistance and E is

voltage


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Physical Origin

1.Newtons Law of Cooling sTTdt

dT

wheredt

dT is rate of cooling of the liquid,

sTT

is temperature difference between the liquid T andits surrounding Ts

dyy

dt

2. Growth and Decay

y is the quantity present at any time

S l ti f Fi t O d Diff ti l


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Solution of First Order DifferentialEquations

Separation of variables

Integrating factor

Substitution methods


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First Order Differential

Equations


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Introduction

Formation of differential equations

Solution of differential equations

First Order Differential Equations


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Introduction

A differential equation is a relationship between an independent

variablex, a dependent variable yand one or more derivatives of ywith

respect tox.

The order of a differential equation is given by the highest derivative

involved. 2

22

2

34

3

0 is an equation of the 1st order

sin 0 is an equation of the 2nd order

0 is an equation of the 3rd orderx

dyx y

dx

d yxy y x

dxd y dy

y edx dx



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Introduction





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Differential equations may be formed from a consideration of the

physical problems to which they refer. Mathematically, they can

occur when arbitrary constants are eliminated from a given function.

For example, let:

2

2

2

2

sin cos so that cos sin therefore

sin cos

That is 0

dyy A x B x A x B x

dx

d yA x B x y

dx

d yy

dx



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Here the given function had twoarbitrary constants:

and the end result was a second orderdifferential equation:

In general an nth order differential equation will result from

consideration of a function with narbitrary constants.

sin cosy A x B x

2

20

d yy

dx



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Introduction





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Introduction

Solving a differential equation is the reverse process to the one just

considered. To solve a differential equation a function has to be

found for which the equation holds true.

The solution will contain a number of arbitrary constantsthe

number equalling the order of the differential equation.

In this Programme, first-order differential equations are considered.



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Direct integration

If the differential equation to be solved can be arranged in the form:

the solution can be found by direct integration. That is:

( )dy

f xdx

( )y f x dx



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Direct integration

For example:

so that:

This is the general solution(orprimitive) of the differential equation.

If a value of yis given for a specific value ofxthen a value for Ccan

be found. This would then be aparticular solutionof the differential

equation.

23 6 5dy

x x

dx

2

3 2

(3 6 5)

3 5

y x x dx

x x x C



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Separating the variables

If a differential equation is of the form:

Then, after some manipulation, the solution can be found by direct

integration.

( )

( )

dy f x

dx F y

( ) ( ) so ( ) ( )F y dy f x dx F y dy f x dx



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Homogeneous equations by substituting y = vx

In a homogeneous differential equation the total degree inxand y

for the terms involved is the same.

For example, in the differential equation:

the terms inxand yare both of degree 1.

To solve this equation requires a change of variable using the

equation:

3

2

dy x y

dx x

( )y v x x



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Examples


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Homogeneous equations by substituting y = vx

To solve:

let

to yield:

That is:

which can now be solved using the separation of variables method.

3

2

dy x y

dx x

( )y v x x

3 1 3 and

2 2

dy dv x y vv x

dx dx x

12

dv vxdx



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Linear equations use of integrating factor

Consider the equation:

Multiply both sides by e5xto give:

then:

That is:

25 xdy

y edx

5 5 5 2 5 75 that isx x x x x xdy d

e e y e e ye edx dx

5 7 5 7

so that

x x x x

d ye e dx ye e C 2 5x xy e Ce



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Linear equations use of integrating factor

The multiplicative factor e5xthat permits the equation to be solved is

called the integrating factorand the method of solution applies to

equations of the form:

The solution is then given as:

where is the integrating factorPdxdy

Py Q edx

.IF .IF where IF Pdx

y Q dx e



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Examples

Substituting

Integrating factor


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Higher Order Differential

Equations


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REDUCTION OF ORDER

Higher Order ODE2ndOrder


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A general second order DE has the form

In this section we consider two special types of second order

equations that can be solved by first order methods.

(1).0)",',,( yyyxF

Reduction of Order

Type A: Dependent variable missing


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When y is not explicitly present, (1) can be written as

Then (2) transforms into

If we can solve (3) for p, then (2) can be solved for y.

)2(.0)",',( yyxf

."and'Let dxdpypy

)3(.0),,( dxdppxf

Type A: Dependent variable missing


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14xyyx The variable y is missing.

let & dpy p ydx

1 , 4dpfrom x p xdx

ExampleSolve the following ODE


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which is linear,

14 (2)

dpp

dx x

1loge

dxxx

IF e e x

4p x x dx c cxcx

px

22

22

4

d


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2dy c

p xdx x

2 c

dy x dxx

dx

x

cxdy

2

22

2 ln2

xy c x c

2 2lny x c x c

l


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Solve the DE.)(

3yyyx

Ans.: .)( 212

2 xccy

Example


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When x is not explicitly present, then (1) can be written as

Then (4) becomes

If we can solve (5) for p, then (4) can be solved for y.

)4(.0)",',( yyyg

dy

dpp

dx

dy

dy

dp

dx

dpypy "then,'Let

( , , ) 0. (5)g y p p dp dy

Type B: Dependent variable missing


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2ln 1 ln lnp y c


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2

1ln 1 lnp c y

1ln 1 ln lnp y c

211 p c y 11

2 ycp

11

2

ycdx

dy 11 yc

dxdy

1( 1)

dy dxc y

1 1 2

2 ( 1)c y c x c

E l


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Find the specified particular solution of the DE

.0when1and

;)'("

21

22

xyy

yyyyy

Ans.:

.832 23x

yey

Example

R d ti f O d


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Reduction of Order

D i ti (1)


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Derivation (1)

D i ti (2)


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Derivation (2)

E l


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Example


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HOMOGENOUS LINEAREQUATIONS WITH CONSTANTCOEFFICIENTS

Higher Order ODE


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Homogenous Linear Equations


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Homogenous Linear Equationswith Constant Coefficients (2)


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Higher Order


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Higher Order


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UNDETERMINED COEFFICIENTSSUPERPOSITION APPROACH

Higher Order ODE

Undetermined Coefficients


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Superposition Approach

Examples


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Examples

Some notes on form of y


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Some notes on form of yp

Trial solutions


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Trial solutions

Examples


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Examples

Examples


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Examples


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VARIATION OF PARAMETER

Higher Order ODE

Variation of Parameters


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Variation of Parameters


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CAUCHY-EULER EQUATION

Higher Order ODE

Cauchy-Euler Equation


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Cauchy-Euler Equation


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Some notes


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Some notes

Examples


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Examples

More Examples


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More Examples

solving system of linear


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equations Simultaneous ordinary differential equations

involve two or more equations that containderivatives of two or more dependentvariables with respect to a singleindependent variable.

The method of systematic elimination forsolving systems of differential equations withconstant coefficient is based on the algebraicprinciple of elimination of variables.

The analogue of multiplying an algebraicequation by a constant is operating on anODE with some combination of derivatives.

Examples


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Examples


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Review: Power series


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Review: Power series

Solution about ordinary point


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Solution about ordinary point

Examples


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Examples


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series solution about regularsingular points (Frobeniuss


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singular points (Frobenius smethod)

Bessels and Legendresti


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equations Bessel functions of first and second kind

Legendres polynomials


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Laplace Transformation

Theory


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Theory

Transforms of Some BasicF nctions


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Functions

Inverse Transform


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Example


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p


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Operational properties


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p p p


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Initial value problems (IVP)


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p ( )

IVPSome Applications


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pp

Free Undamped Oscillation


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p


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Free Damped Oscillation


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p


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END OF COURSE

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