Page 231 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes classic problem of Cauchy, which in the case of two independent variables may be stated as
follows:

Cauchy’s Problem

Cauchy’s problem is stated as follows:

z z
(a) x(t), y(t), and z(t) are functions which together with their first derivatives , are
x y
continuous in the interval M defined by t < t < t ,
1 2
z z z z
(b) And if (x, y, z, , ) is continuous function of x, y, z, p = ,q in a certain region
x y x y
U of the xyz pq space, then it is required to establish the existence of the function z = f(x, y)
with the following properties:
(1) f(x, y) and its partial derivatives with respect to x and y are continuous functions of x
and y in a region R of the xy space.
(2) For all values of x and y lying in R the point {x, y, f(x, y), f (x, y), f (x, y)} lies in U and
x y
[x, y, f(x, y), f (x, y), f (x, y)] = 0
x y
(3) For all t belonging to the interval M, the point {x (t), y (t)} belongs to the region R
0 0
and
f{x (t), y (t)} = z
0 0 0
Geometrically stated, what we wish to prove is that there exists a surface z = f(x, y) which
passes through the curve whose parametric equations are
x = x (t), y = y (t) and z = z (t) ...(1)
0 0 0
and at every point of which the direction (p, q, 1) of the normal is such that

{x, y, z, p, q} = 0 ...(2)
The Cauchy’s problem stated above can be formulated in seven other ways. For details you are
referred to D. Berstein. To prove the existence of a solution it is necessary to make some more
assumptions about the form of the functions and the curve. There are a whole class of existence
theorems depending on the nature of these assumptions. However we shall be contented our-
selves by quoting one of them as follows.
Theorem: If g(y) and all its derivatives are continuous for |y y | < , if x is a given number and
0 0
z = g(y ), q = g (y ) and if (x, y, z, q) and all its partial derivatives are continuous in a region S
0 0 0 0
defined by
|x x |< , |y y |< , |q q |<
0 0 0
then there exists a unique function (x, y) such that:

(a) (x, y) and all its partial derivatives are continuous in a region R defined by |x x |< ,
0 1
|y y | < .
0 2
(b) For all (x, y) in R, z = (x, y) is a solution of the equation
z z
= f(x, y, z, )
x y
(c) For all values of y in the interval |y y | < , (x , y) = g(y).
0 1 0

224 LOVELY PROFESSIONAL UNIVERSITY