Page 200 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 12: Sturm Comparison and Separation Theorems

where P(x), Q(x) and R(x) are finite polynomials that contain no common factor. This equation is Notes
inhomogeneous and has variable coefficients. After dividing through P(x), we obtain the more
concurrent, equivalent form,
2
d y dy
a 1 ( ) a 0 ( )y f ( ) ...(1)
x
x
x
dx 2 dx
Provided p 0. If p(x) = 0 at some point x = x , we call x = x a singular point of the equation. If P(x)
0 0
0, x is a regular or ordinary point of the equation. If P(x) 0 for all points x in the interval
0
where we want to solve the equation, we say the equation is non-singular or regular in the
interval.
If a (x), a (x) and f(x) are continuous on some open interval a < x < b that contains the initial point,
1 0
then a unique solution of the form
y = Au (x) + Bu (x) + G(x)
1 2
where A, B are constants and are fixed by initial conditions. Before we try to construct the
general solution of equation (1), we will outline a series of sub-problems that are more tractable.

12.2 The Method of Reduction of Order

As a first simplification we discuss the solution of the homogeneous differential equation

2
d y dy
x
a 1 ( ) a 0 ( )y 0 ...(2)
x
dx 2 dx
on the assumption that we know one solution, say y(x) = u (x), and only need to find the second
1
solution. We will look for a solution of the form y(x) = U(x)u (x). Differentiating y(x) using the
1
product rule gives
dy dU du 1
u 1 U ,
dx dx dx
2
2
2
d y d U u 2 dU du 1 U d u 1
dx 2 dx 2 1 dx dx dx 2
If we substitute these expressions into (2) we obtain
2
2
d U u 2 dU du 1 U d u 1 a ( ) dU u U du 1 a ( )Uu 0
x
x
dx 2 1 dx dx dx 2 1 dx 1 dx 0 1
We can now collect terms to get
2
2
d u du d U dU du
x
U 1 a 1 ( ) 1 a 0 ( )u 1 u 1 2 1 a u 0
x
1 1
dx 2 dx dx 2 dx dx
Now, since u (x) is a solution of (2), the term multiplying U is zero. We have therefore obtained
1
a differential equation for dU/dx, and, by defining Z = dU/dx, we have
dZ du
u Z 2 1 a u 0
1 1 1
dx dx
Dividing through by Zu we have
1
1 dZ 2 du 1 a 0,
Z dx u dx 1
1

LOVELY PROFESSIONAL UNIVERSITY 193