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Homework Statement
Determine if the given set of functions is linearly independent or linearly dependent.
Homework Equations
$$S=x~sin~x, ~ x~cos~x$$
The Attempt at a Solution
My first instinct was to use the Wronskian.
$$W[y_1(x), y_2(x)]=\begin{vmatrix}
x~sin~x & x~cos~x\\
x~cos~x+sin~x & cos~x-x~sin~x
\end{vmatrix}$$
Now I take the determinant and I get -
$$W=x~cos~x(cos~x-x~sin~x)-x~cos~x(x~cos~x+sin~x)$$
$$W=x~cos~x~sin~x-x^2~sin^2~x-x^2~cos^2~x-x~cos~x~sin~x$$
$$W=-x^2~sin^2~x-x^2~cos^2~x$$
$$W=-x^2(sin^2~x+cos^2~x)$$
$$W=-x^2$$
Now, since ##-x^2\neq0##, can I conclude that the functions are linearly independent?
I've done some examples in my textbook that involved determining linear independence on the interval of all real numbers, and there were some instances when the Wronskian didn't equal 0, but the answer key said that they were linearly dependent. I understood that the Wronksian was to determine linear independence or dependence. Am I correct, or do I misunderstand the Wronksian?
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