Exercise 25

We define two functions, $f(t)$ and $g(t)$, to be linearly independent on an open interval $I = (a, b)$ if there do not exist nonzero constants $c_1$ and $c_2$ such that $$c_1 f(t) + c_2 g(t) = 0$$ for all $t \in I$. Equivalently, two functions are linearly independent, if one function is not a multiple of the other. Otherwise, $f(t)$ and $f(t)$ are linearly dependent. Suppose that $f(t)$ and $g(t)$ are solutions to the homogeneous linear equation $$y'' + p y' + q y = 0.$$ Show that $f(t)$ and $g(t)$ are linearly dependent on an interval $I = (a, b)$ if and only if $W[f, g](t) \equiv 0$, where $W[f, g](t) = f(t) g'(t) - g(t) f'(t)$ is the Wronskian of $f$ and $g$.