Is the set of functions {f_1(x) = 1, f_2(x) = sin^2 x, f_3(x) = cos^2 x) linearl

Question

Is the set of functions {f_1(x) = 1, f_2(x) = sin^2 x, f_3(x) = cos^2 x) linearly dependent or linearly independent on (-infinity, infinity) One solution of the differential equation y'' - y' = 0 is y = e^x. Use the method of reduction of order to find a second linearly independent solution. Solve the differential equation y" + 3y' + 2y = 0. Solve the differential equation y" + 4y' + 4y = 0. Solve the differential equation y" + 4y' + 5y = 0. Solve the differential equation y'' + 3y' + 2y = 3x + 1. Solve the differential equation y" + 4y' + 4y = cos(2x). Solve the differential equation y" + 4y' + 5y = 2xe^x. Solve the differential equation y" + 3y' = 4x - 3. Solve the differential equation y" + 3y' + 2y = 4e^-x. Solve the differential equation y" + 4y' + 4y = e^-2x. Without solving for the undetermined coefficients, what is the correct form of the particular solution of the differential equation y'' + 4y' + 5y = e^-2t cos x

Explanation / Answer

Hi, first of all, you have to post each question in a unique post. I'm going to answer the first, since you didn't specified any.
We want to know if the set of functions
f1(x) = 1,
f2(x) = sin^2(x),
f3(x) = cos^2(x),
is or not linearly independent in the interval (-infinity, +infinity).
The set of functions would be linearly independent if we cannot write one of the functions in terms of the others.
However, we know that there exists a trigonometrical identity that says that sin^2(x)+cos^2(x) = 1.
This written in terms of the given functions looks as
f1(x) = f2(x) + f3(x).
Since we were able to write f1 in terms of f2 and f3, therefore the set is linearly dependent.

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