= Solution
The <Wronskian> is
$$
\boxed{W(f,g)(x)=\begin{vmatrix}f(x)&g(x)\\f'(x)&g'(x)\end{vmatrix}=f(x)g'(x)-f'(x)g(x).}
$$
Suppose constants $A,B$ satisfy $Af+Bg=0$ throughout the interval. Differentiating gives $Af'+Bg'=0$. At $x_0$, these form the homogeneous linear system
$$
\begin{pmatrix}f(x_0)&g(x_0)\\f'(x_0)&g'(x_0)\end{pmatrix}\begin{pmatrix}A\\B\end{pmatrix}=0.
$$
Its <determinant> is nonzero by hypothesis, so $A=B=0$. This is exactly <linear independence> of the two functions on the interval. No differential equation is needed for this implication; the converse for arbitrary differentiable functions would require additional hypotheses.
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