Determining if the functions {cosx , e^-x , x} are linearly independent

[chris_0101]

Homework Statement

Basically, the title says it all, I need to figure out whether these functions are linearly independtend on (-infinity, infinity)

Homework Equations

Wronskian (the determinant of the matrix composed of the functions in the first row, first derivative in the second row and second derivatives in the third row)

The Attempt at a Solution

After computing the Wronskian this is what I got:
[(-e^-x)(-cosx)] + [(xe^-x)(-sinx)] - [(x)(-e^-x)(cosx)] - [(e^-x)(cosx)]

however, I cannot seem to simply this. If anyone can help me simplify this further that would be great. Also if you help me determine if whether they are linearly independent.

Thanks

 

[HallsofIvy]

Okay, you have calculated the Wronskian. Why? What does the Wronskian tell you? Would it help you to observe that, if $x= \pi/2$, that reduces to $-(\pi/2)e^{-\pi/2}$?

 

[chris_0101]

basically the wronskian tells us that if it is not equal to zero the specified functions are linearly independent.

After evaluating what you told me to substitute in, I get

(-1/2)(e^(-pi/2))(pi)

With this, the wronskian can never equal to zero due to the function of e.

Is this correct?