Maths Tm Paper

7/30/2019 Maths Tm Paper

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In mathematics, an ordinary differential equation (ODE) is

a relation that contains functions of only one independentvariable, and one or more of their derivatives with respect to

that variable.

A simple example is Newton's second law of motion, which

leads to the differential equation

for the motion of a particle of constant mass m. In general,

the force F depends upon the position x(t) of the particle at

time t, and thus the unknown function x(t) appears on both

sides of the differential equation, as is indicated in the

notation F(x(t)).Ordinary differential equations are

distinguished from partial differential equations, which

involve partial derivatives of functions of several variables.

Ordinary differential equations arise

in many different contexts including geometry, mechanics,

astronomy and population modelling. Many famous

mathematicians have studied differential equations and

contributed to the field, including Newton, Leibniz, the

Bernoulli family, Riccati, Clairaut, d'Alembert and Euler.


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1. Existence and Uniqueness

2. Defination

3. Examples

4. Reduction to a first order system

5. Linear order differential equation

6. Theories of ODEs

7. References


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Consider a first order differential equation

with the initial condition

where is bounded in the neighbourhood of the initial

point, i.e.,

Sufficient Condition of Existence: If is continuous in

the neighbourhood region , the solution of thisinitial value problem in the region exists.

Sufficient Condition of Existence and Uniqueness: If

and its partial derivative with respect to are continuous in

the neighbourhood region , the solution of thisinitial value problem in the region exists and is unique.

Picard Iteration Method: The unique solution of the above

initial value problem is

where


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Ordinary differential equation

Let y be an unknown function

in x with y(n) the nth derivative of y, and let F be a

given function

then an equation of the form

is called an ordinary differential equation (ODE) of

order n. If y is an unknown vector valued function

it is called a system of ordinary differential equations

of dimension m (in this case, F : mn+1 m).

More generally, an implicit ordinary differential

equation of order n has the form


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where F : n+2 depends on y(n). To distinguish

the above case from this one, an equation of the form

is called an explicit differential equation.

A differential equation not depending on x is

called autonomous.

A differential equation is said to be linear if F can bewritten as a linear combination of the derivatives of y

together with a constant term, all possibly depending

on x:

with ai(x) and r(x) continuous functions in x. The

function r(x) is called the source term; if r(x)=0 then

the linear differential equation is called

homogeneous, otherwise it is called non-

homogeneous or inhomogeneous.

Solutions

Given a differential equation


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a function u: I R R is called the solution or

integral curve for F, if u is n-times differentiable on I,

and

Given two solutions u: J R R and v: I R

R, u is called an extension of v if I J and

A solution which has no extension is called a global

solution.

A general solution of an n-th order

equation is a solution containing n arbitraryvariables, corresponding to n constants of integration.

A particular solution is derived from the general

solution by setting the constants to particular values,

often chosen to fulfill set 'initial conditions or

boundary conditions'. A singular solution is a solution

that can't be derived from the general solution.


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Any differential equation of order n can be written as

a system of n first-order differential equations. Given

an explicit ordinary differential equation of order n.

define a new family of unknown functions

for i from 1 to n.

The original differential equation can be rewritten as

the system of differential equations with order 1 and

dimension n given by

which can be written concisely in vector notation as

with

and


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A well understood particular class of differential

equations is linear differential equations. We can

always reduce an explicit linear differential equation

of any order to a system of differential equation of

order 1

which we can write concisely using matrix and vector

notation as

with


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can be constructed by finding the fundamental system

to the corresponding homogeneous equation and one

particular solution to the inhomogeneous

equation. Every solution to nonhomogeneous

equation can then be written as

A particular solution to the nonhomogeneous

equation can be found by the method of undetermined

coefficients or the method of variation of parameters.

Concerning second order linear ordinary differentialequations, it is well known that

So, if yh is a solution of: y'' + Py' + Qy = 0 , then

such that : Q = s' s2 Sp.So, if yh is a solution of: y'' + Py' + Qy = 0 ; then a

particular solution yp of y'' + Py' + Qy = W , is given

by:


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Fundamental systems for homogeneous equations

with constant coefficients

If a system of homogeneous linear differentialequations has constant coefficients

then we can explicitly construct a fundamentalsystem. The fundamental system can be written as a

matrix differential equation

with solution as a matrix exponential

which is a fundamental matrix for the original

differential equation. To explicitly calculate this

expression we first transform A into Jordan normal

form

and then evaluate the Jordan blocks


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of J separately as

The theory of singular solutions of ordinary and

partial differential equations was a subject of research

from the time of Leibniz, but only since the middle of

the nineteenth century did it receive special attention.

A valuable but little-known work on the subject is that

of Houtain . Darboux was a leader in the theory, and

in the geometric interpretation of these solutions he

opened a field which was worked by various writers,

notably Casorati and Cayley.


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The primitive attempt in dealing with differential

equations had in view a reduction to quadratures. As

it had been the hope of eighteenth-century algebraists

to find a method for solving the general equation of

the nth degree, so it was the hope of analysts to find a

general method for integrating any differential

equation. Gauss (1799) showed, however, that the

differential equation meets its limitations very soon

unless complex numbers are introduced. Henceanalysts began to substitute the study of functions,

thus opening a new and fertile field. Cauchy was the

first to appreciate the importance of this view.

Thereafter the real question was to be, not whether a

solution is possible by means of known functions or

their integrals, but whether a given differential

equation suffices for the definition of a function of

the independent variable or variables, and if so, what

are the characteristic properties of this function.

Two memoirs by Fuchs, inspired a novel approach,

subsequently elaborated by Thom and Frobenius.

Collet was a prominent contributor beginning in 1869,

although his method for integrating a non-linear

system was communicated to Bertrand in 1868.


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infinitesimal transformations of solutions to solutions

(Lie theory). Continuous group theory, Lie algebras

and differential geometry are used to understand the

structure of linear and nonlinear (partial) differential

equations for generating integrable equations, to find

its Lax pairs, recursion operators, Bcklund

transform and finally finding exact analytic solutions

to the DE.

Symmetry methods have been recognized to study

differential equations arising in mathematics, physics,

engineering, and many other disciplines.

SturmLiouville theory is a theory of eigenvalues and

eigenfunctions of linear operators defined in terms of

second-order homogeneous linear equations, and is

useful in the analysis of certain partial differential

equations.


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My project on superconducting quantam interference

devices is made with the help of google site,

Wikipedia, using text book, my own knowledge and

with the help of my co-teachers who made it possible

for me to make this project easily. I am very thankful

to my teachers for their help in making my project.

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Maths Tm Paper