Solutions Of De (08 Ee 09)

SOLUTIONS OF

DIFFERENTIAL EQUATIONS

ASAD ABBAS08-EE-09

Solutions Of Differential Equations 2


4 2 2 3sin , ' 2 0, 0y x y y xy x y y x

Definition A differential equation is an equation involving derivatives of an unknown function and possibly the function itself as well as the independent variable.

Example

Definition The order of a differential equation is the highest order of the derivatives of the unknown function appearing in the equation

1st order equations 2nd order equation

sin cosy x y x CExamples 2 3

1 1 26 e 3 e ex x xy x y x C y x C x C

In the simplest cases, equations may be solved by direct integration.

Observe that the set of solutions to the above 1st order equation has 1 parameter, while the solutions to the above 2nd order equation depend on two parameters.


WHY STUDY DIFFERENTIAL EQUATIONS?

• Some examples of fields using differentialequations in their analysis include:– solid mechanics & motion– heat transfer & energy balances– vibrational dynamics & seismology– aerodynamics & fluid dynamics– electronics & circuit design– population dynamics & biological systems– climatology and environmental analysis– options trading & economics


INTERESTINGAPPLICATIONS

OF DIFFERENTIAL EQUATIONS


Chaos in the brain Saying that someone is a chaotic thinker might seem like an insult but it could be that the mathematical phenomenon of chaos is a crucial part of what makes our brains work. Chaos is all about unpredictable change and this can be described using differential equations.

How the leopard got its spots How does the uniform ball of cells that make up an embryo differentiate to create the dramatic patterns of a zebra or leopard? How come there are spotty animals with stripy tails, but no stripy animals with spotty tails? Get to know the equations that explain all this and more.

Going with the flow This article describes what happens when two fluids of different densities meet, for example when volcanoes erupt and hot ash−laden air is poured out into the atmosphere. The article explains Newton's second law of motion as a differential equation and its relation to fluid mechanics.

How plants halt sands Plants can stop the desert from relentlessly invading fertile territory. But just how and where should they be planted? A model involving differential equations gives the answers.


In Physics and Technology:

Light attenuation and exponential laws Many natural processes adhere to exponential laws. The attenuation of light the way it decays in brightness as it passes through a thin medium is one of them. The article explores the attenuation law of light transmission in its differential form.

Computer games developer In the real world, balls bounce and water splashes because of the laws of physics. In computer games, a physics engine ensures the virtual world behaves realistically. Nick Grey explains that to make the games, you need to understand the physics, and that requires differential equations.

Spaghetti breakthrough Differential equations model the breaking behavior of pasta.

Fluid mechanics researcher Trying to solve differential equations can give you a stomach ache sometimes, but the equations can also help to prevent one. André Léger uses fluid dynamics to understand how food sloshes around the intestines.


In Sports:

If you can't bend it, model it! David Beckham and his fellow players may intuitively know how to bend a football's flight as they wish, but the rest of us have to resort to the differential equations describing the aerodynamics of footballs

Aerodynamicist The smallest alteration in the shape of a Formula One car can make the difference between winning and losing. It's the air flow that does it, so, as Christine Hogan explains, any Formula One team needs an aerodynamicist.

Formulaic football Mathematicians build a mathematical model of a football match.


In Finance:

Financial modelling David Spaughton and Anton Merlushkin work for Credit Suisse First Boston, where they provide traders in the hectic dealing room with software based on complicated mathematical models of the financial markets. They explain how changing markets need the maths of change.

Financial maths course director Riaz Ahmad's mathematical career has led him from the complexities of blood flow to the risks of the financial markets via underwater acoustics differential equations help to understand all of these.

Project Finance Consultant Nick Crawley set up his own financial consultancy firm in Sydney, Australia, offering advice on large−scale financing deals. Understanding the risks of investments means understanding the fluctuations of markets, and that requires differential equations.


SOLUTIONS OF



Analytical Solutions:

An analytical representation of a solution may take one of two forms:

1. In the explicit form y=f(t), the dependent variable is completely isolated and appears only to the first power on one side of the equation. The other side of the equation is an expression involving only the independent variable t and constants.

y(t) = Ce2t ‘is an explicit solution, but y(t) =ty

C

is not.

2. The implicit form is an equation h(t,y) = 0 involving both the dependent and independent variables but no derivatives. In this form the dependent variable y is not expressly given as a function of the independent variable t. We assume that the implicit form is satisfied by at least one function that also satisfies the differential equation. The equation cyt 22 represents an implicit solution to the differential equation =

y

t

dt

dy


In general the analytic solution of a first-order o.d.e. will have one arbitrary constant, the solution of a second-order o.d.e. will have two arbitrary constants, while the solution of an nthorder o.d.e. will have n arbitrary constants. We refer to these as one-parameter, two- parameter, or n-parameter solutions. Because the arbitrary constants (parameters) can take on infinitely many values, these one-, two-, or n-parameter solutions represent families of solutions.

A general solution to an nth-order o.d.e. is an n-parameter analytic solution (expressed explicitly or implicitly) that contains all possible solutions over an interval I. All linear nth-order o.d.e.s have general solutions.


Graphical Solution:

A graphical Solution of first order differential Equation is the curve whose slope at any point is the value of the derivative there as given by the differential equation.

Graphical solutions may be quantitative in nature; i.e., the graph may be sufficiently precise so that the values of the solution function can be read directly from the graph.

Graphical solutions may be qualitative in nature where the graph is imprecise as far as numerical values are concerned yet still revealing of the general shape and features of the solution curves.

Graphical solutions can be produced in different ways: from a table of numerical values, by plotting an analytic solution, or by using a direction or tangent field of’ the differential equation.

Numerical Approximations: A solution to a differential equation may also be approximated numerically. In this case the form of the solution is a sequence or table of values of the dependent variable y for a preselected sequence of values of the independent variable 1.


Example:

Find the general solution to

xxdx

dy2

1

•

•

•

dxxx

dy

2

1

dxx

xy 2

1

Cxxy 2ln


INITIAL CONDITIONSINITIAL VALUE PROBLEM

ANDBOUNDARY CONDITIONS


• In many physical problems we need to find the particular solution that satisfies a condition of the form y(x0)=y0. This is called an initial condition, and the problem of finding a solution of the differential equation that satisfies the initial condition is called an initial-value problem.

• Example :

y2 = x2 + C satisfying the initial condition y(0) = 2. 22 = 02 + CC = 4y2 = x2 + 4

Cxy

xdxydyy

x

dx

dy

22


Initial Value Problems:

An nth-order Initial Value Problem (LV.P.) consists of an nth-ordcr o.d.e. )1()( ,......,'',',,( nn yyyytfy

together with n initial conditions,

)1()1( )(,.......,'')('',')(' noo

noooo ytyytyyty

The solution of an nth-order I.V.P. is a specific solution where the arbitrary constants has been assigned number values in order to satisfy the initial conditions. This solution must be continuous on an interval containing the initial t value, to. and must have the value yo at to.


Example:

Find the particular solution to

0 when 1 ;23 xyxxdx

dy

3 3 4 212 2

4dy x x dx y x x dx x x C

4 211 0 0 1

4C C

14

1 24 xxy

Apply the initial condition: y(0) = 1

•

•


Boundary Condition:

It is a condition on the solution at two or more points or it is a pair of corresponding value. Or Conditions given at the boundary of some region.

Boundary Value Problem:

An Ordinary Differential Equation where not all of the data Is given at one point is an Boundary Value Problem. For example, y - y = 0 with the data y(0) = 1, y (1) = 1 is a Boundary Value Problem.


Solved ProblemExercise 1.1Page no. 8

Question no. 10


Problem: 10What Happens to DE in Problem 9 if we change the

solution to

1

:

0'

22

yx

solution

yyx

122 yx

Problem: 9


0'

0

0)(2

022

0

1

22

22

yyxdx

dyyx

dx

dyyx

dx

dyyx

dx

dy

dx

dx

Solution

yx

(Diff w.r.t ‘x’)

Ans.

Solutions Of De (08 Ee 09)

Education

Transcript of Solutions Of De (08 Ee 09)