Page 109 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes dy d y d y
n
2
y
 x , , , 2 ..... n 0 ...(1)
dx dx dx
The general solution of an nth order differential equation involves n arbitrary constants a , a ,
1 2
..... a . In the following we shall study the existence of an ordinary first order differential equation.
n
The ordinary differential equation of the first order is generally written in the form
dy
y
 x , , 0 ...(2)
dx
we shall study the solution of the equation (2) with the initial conditions i.e. at
x = x , y = y ...(3)
0 0
We can vary x in a certain range i.e.
x h x x + h ...(4)
0 0
where h is an increment to x. The above range of x is in a domain D. When x varies in the above
range we want to see how y changes from the initial value y . Let us assume that y varies in the
0
range
y k y y + k ...(5)
0 0
So let D be a domain in (x, y) plane given by (4) and (5). Let the set of points in (4) are given by
x , x , ... x ,... and set of points in (5) are given by y , y , ... y ,... . We want to study the existence
0 1 n 0 1 n
and uniqueness of the solution of equation (2). There are various forms of (2). We in particular
study the equation in the form
dy
= f(x, y) ...(6)
dx
subject to the initial conditions (3).

6.2 Picard’s Method

Our purpose is to find a solution of equation (6) subject to the initial condition (3). To formula
the problem we have to make the following assumptions concerning f(x, y). The behaviour of
f(x, y) will decide the solution of (6).

Assumption 1: The function f(x, y) is real-valued and continuous on a domain D of the (x, y) plane
given by
x h x x + h, y k y y + k ...(7)
0 0 0 0
Here h, k are positive numbers.
Assumption 2: f(x, y) satisfies the Lipschitz condition with respect to y in D, that is, there exists a
positive constant k such that

|f(x, y ) f(x, y )| k|y y | ...(8)
1 2 1 2
for every pair of points (x, y ), (x, y ) of D.
1 2
x
t  ( , )
y
If f(x, y) has a continuous partial derivative then assumption 2 is satisfied. Now since D
y 
y
x
x
f ( , ) f ( , )
y
is a bounded closed domain and is continuous in D so is bounded. Put
y y
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