Fact of the wronskian method, Mathematics

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Given two functions f(x) and g(x) which are differentiable on some interval I

 (1) If W (f,g) (x0) ≠ 0 for some x0 in I, so f(x) and g(x) are linearly independent on the interval I.

 (2) If f(x) and g(x) are linearly dependent on I so W(f,g)(x) = 0 for all x in the interval I.

Be very alert with this fact. This DOES NOT say as if W(f,g)(x) = 0 so f(x) and g(x) are linearly dependent! Actually it is possible for two linearly independent functions have a zero Wronskian!

This fact is utilized to quickly know linearly independent functions and functions which are liable to be linearly dependent.

Example 2: Verify the fact by using the functions,

a) f x ) = 9 cos ( 2x )      g x ) = 2 cos2 ( x ) -  2 sin 2 ( x )

(b) f (t ) = 2t 2                                g (t ) = t 4

Solution:

(a) f x ) = 9 cos ( 2x )     g x ) = 2 cos2 ( x ) -  2 sin 2 ( x )

In this case if we calculate the Wronskian of the two functions we must get zero as we have already found that these functions are linearly dependent.

766_Fact of the wronskian method.png

Therefore, we get zero as we must have. See the heavy use of trig formulas to simplify the work!

(b) f (t ) = 2t 2                                g (t ) = t 4

Now there we know that the two functions are linearly independent and therefore we should find a non-zero Wronskian.

1474_Fact of the wronskian method1.png

= 8t5 - 4t5 = 4t5

The Wronskian is non-zero as we estimated provided t ≠ 0. It is not a problem. As long as the Wronskian is not as identically zero for each t we are okay.


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