Existence and Uniqueness Theorems for FirstOrder ODEs
The general rstorder ODE is
y = F(x, y), 31(900) = y0- (*)
We are interested in the following questions:
(i) Under what conditions can we be sure that a solution
to (*) exists?
(ii) Under what conditio
1
Vector Spaces
1.1
Introduction
The notion of a vector space is an abstraction of the familiar set of vectors in two or three
dimensional Euclidian space. For example, let ~x = (x1 , x2 ) and ~y = (y1 , y2 ) be two vectors in
the plane R2 . Then we have
6
Vector Spaces
1.3
Subspaces
We observe that
V = cfw_x = (x1 , x2 ) R2 : x2 = 0, which is a subset of R2 is a vector space with respect to
the addition and scalar multiplication as in R2 .
V = cfw_x = (x1 , x2 ) R2 : 2x1 + 3x2 = 0 which is a subset of
Basis
1.6
11
Basis
Definition 1.10 (Basis) A subset E of a vector space V is said to be a basis of V if it is
linearly independent and span E = V .
EXAMPLE 1.18 For each j cfw_1, . . . , n, let ej Fn be such that ej (i) = ij , i, j = 1, . . . , n.
Then we
MA 2030
Linear Algebra and Numerical Analysis
Arindama Singh & S. Mishra
Prepared by
A. V. Jayanthan & Arindama Singh
AS & SM 0
Lecture slides, topic wise as we progress, will be available in
http:/mat.iitm.ac.in/home/asingh/public_html/teaching.html
It w
7
Greens Functions and Nonhomogeneous
Problems
The young theoretical physicists of a generation or two earlier subscribed to the
belief that: If you havent done something important by age 30, you never will.
Obviously, they were unfamiliar with the histor
NOTES ON THE EXISTENCE AND UNIQUENESS THEOREM
FOR FIRST ORDER DIFFERENTIAL EQUATIONS
I. Statement of the theorem.
We consider the initial value problem
cfw_ y(w) = F(w,y($)
y($o) = yo.
(1.1)
Here we assume that F is a function of the two variables (x, y),
Chapter 5
Boundary Value Problems
A boundary value problem for a given differential equation consists of finding a solution of the
given differential equation subject to a given set of boundary conditions. A boundary condition
is a prescription some combi
S. Ghorai 1
Lecture V
Picards existence and uniquness theorem, Picards iteration
1 Existence and uniqueness theorem
Here we concentrate on the solution of the rst order IVP
1/ = J00, 1/); 21(900) = 3/0 (1)
We are interested in the following questions:
1.
Linear Algebra and Numerical Analysis
Assignment-1
Jan-May, 2012
In the following V denotes a vector space over F which is R or C, and Fmn denotes the
set of all m n matrices with entries from F.
1. For x, y V , show that x + y = x implies y = 0.
2. Let x