SOLUTION: Find the Wronskian of 1, cos2x, sin²x at x=PI/2. Are these functions dependent?
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Question 607237: Find the Wronskian of 1, cos2x, sin²x at x=PI/2. Are these functions dependent?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
General 3X3 Wronskian:
For your functions:
\ \ \ \ \ \sin^2(x)\cr 0\ \ \ -2\sin(2x)\ \ \ 2\cos(x)\sin(x)\cr 0\ \ \ -4\cos(2x)\ \ \ 2\cos^2(x)-2\sin^2(x)\right|)
At 
:
Verification of the value of the discriminant left as an exercise for the student. The Wronskian is zero, so the system could be dependent, but not necessasarily.
If they are linearly dependent, then you should be able to find constants 
, two of which must be non-zero such that:
\ +\ c_3\sin^2(x)\ =\ 0)
for all 
. Intuitively speaking, I'd say they are independent.
John
My calculator said it, I believe it, that settles it
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