Linearly independence and dependence of functions
And application of Wronskian:
In this lecture we study how the linearly independence or dependence of two or more
functions is checked. This is not an easy topic. You need some good mathematical
knowledge but it is very interesting that we make it easy. Do not worry only walk with
me. We also study the application of Wronskian. Before studying the wronskian, we need
to introduce couple of new concepts first.
Given two non-zero functions and write down the following equation
Note:
If , then is true for all values of , regardless of the functions that we
use. Consequently the functions should be linearly independent.
But if we are able to find non-zero constants and for which will also be
true for all values of , then the two functions must be linearly dependent.
Hence the values of the constants determine the linearly independence and
dependence of functions. We elaborate this concept taking few examples.
Example:
Determine the linearly independence and dependence of the following
functions:
and
Solution:
To check the linearly independence and dependence of the given functions,
we will determine the values of the constants and , if they are non-zero then the
functions are linearly dependent. If both are equal to zero then the functions are
linearly independent. So, is written as;
Comparing the coefficients of like terms of the left hand sides to the right hand sides,
we have the following information:
Adding and , we have
Putting the value of k = 0 in (a), we have h = 0, hence (2) is only satisfied for
, this shows that the above given functions are linearly independent.
Wronskian is a very good tool to check the linearly independence of the
functions if the wronskian of the functions is non-zero for at least on value of ,
then the function would be linearly independent.
Example:
Determine the linearly independence of the following functions;
and
Solution:
This shows that the functions are linearly independent.
Example:
Determine the linearly independence and dependence of the following
functions:
and
Solution:
We need to determine if we can find non-zero constants h and k that will make this true
for all x or if k = 0 and h = 0 are the only constants that will make this true for all x. This
is often a fairly difficult process. The process can be simplified with a good intuition for
this kind of thing, but that’s hard to come by, especially if you haven’t done many of
these kinds of problems. Now, we try to write in terms of , as;
Now can be written as;
Above is equal to zero for infinite values of h and k, we will calculate some values.
Above all non-zero values of the constants h and k satisfies , this shows that the given
functions are linearly dependent.
Exercise NO. 02:
Complete the following table:
Type
Homogeneous/
No Functions Reason
Non
Homogeneous
1
2
3
4
5
Exercise NO. 03:
Complete the following table:
Type
Degree Degree
Homogeneous/
No Differential Equations Reason of of
Non
M(x,y) N(x,y)
Homogeneous
1
