Question 607237
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General 3X3 Wronskian:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left|f\ \ \ \ \,g\ \ \ \ \,h\cr f'\ \ \ g'\ \ \ h'\cr f^{[2]}\  g^{[2]}\ h^{[2]}\right|]


For your functions:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left|1\ \ \ \ \,\cos(2x)\ \ \ \ \ \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 *[tex \LARGE x\ =\ \frac{\pi}{2}]:



*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left|1\ -1\ \ \ 1\cr 0\ \ \,0\ \ \ 0\cr 0\ \ \,4\ -2\right|\ =\ 0]


Verification of the value of the discriminant left as an exercise for the student.  The Wronskian is zero, so the system <i><b>could be</b></i> dependent, but not necessasarily.


If they are linearly dependent, then you should be able to find constants *[tex \LARGE c_1,\ c_2,\ c_3], two of which must be non-zero such that:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ c_1\ +\ c_2\cos(2x)\ +\ c_3\sin^2(x)\ =\ 0]


for all *[tex \Large x].  Intuitively speaking, I'd say they are independent.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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