the Wronskian
/ RON-skee-an /
Suppose someone hands you two solutions of a second-order equation and asks: are these two genuinely different, enough to build every solution from? Checking the ratio by eye can be awkward, especially if the functions are messy. The Wronskian is a single number you can compute at a point that answers the question with certainty.
For two functions y1 and y2 the Wronskian is the determinant of the matrix whose rows are the functions and their first derivatives: W(y1, y2)(x) = y1 y2' - y2 y1'. You evaluate it like any 2-by-2 determinant — top-left times bottom-right minus top-right times bottom-left. The key fact, for two solutions of the same linear equation y'' + p y' + q y = 0, is this dichotomy: either W is zero at every point of the interval (and the solutions are linearly dependent), or W is never zero on the interval (and they are independent). There is no in-between, no patch where it dips through zero and comes back. So you may test at any single convenient point x0: if W(x0) is not zero, the pair is independent and forms a fundamental set.
The Wronskian is the practical certificate of independence, and it does more. It tells you whether two given solutions can match arbitrary initial conditions y(x0) and y'(x0) — they can exactly when W(x0) is nonzero, because W is precisely the determinant of the linear system you solve for c1 and c2. It feeds the variation-of-parameters formula for forced equations. And it obeys Abel's identity, a differential equation of its own that lets you find W without ever solving the original equation.
For e^(2x) and e^(-3x), W = e^(2x)(-3 e^(-3x)) - e^(-3x)(2 e^(2x)) = -3 e^(-x) - 2 e^(-x) = -5 e^(-x), which is never zero. So this pair is independent everywhere and forms a fundamental set for the equation it solves.
A single evaluation of the Wronskian — here clearly nonzero — certifies independence on the whole interval.
The all-or-nothing dichotomy (zero everywhere or nowhere) holds only for solutions of the same linear ODE. For arbitrary functions the Wronskian can vanish at isolated points or be zero while the functions are still independent, so do not over-trust it outside that setting.