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When proving this, my textbook first writes down the determinant, and then states:

This will be tedious to compute directly, and we only need to know that the Wronskian is nonzero at a single point, so we evaluate it at $x = 0$

Why is this?

I recall that there's a similar theorem that states if these functions are solutions to a differential equation

then they are linearly independent $\iff$ Wronskian = 0 at one point in the interval $\iff$ Wronskian = 0 at all points in the interval.

However, the problem in the header does not state that they are solutions to any differential equation.

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