Wronskian
/ VRON-skee-an /
Suppose you have two candidate solutions and want a single number that tells you whether they are genuinely different or secretly the same up to scaling. The Wronskian is that number — a determinant built from the functions and their derivatives — and it works as an independence detector for solutions of linear differential equations.
For functions y_1, ..., y_n, the Wronskian W(x) is the determinant of the matrix whose first row is the functions themselves, second row is their first derivatives, and so on down to the (n-1)-th derivatives. For two functions it is simply W = y_1 y_2' - y_2 y_1'. If y_1 and y_2 are solutions of the same second-order linear ODE y'' + p(x) y' + q(x) y = 0, then Abel's identity says W(x) = W(x_0) exp(-integral of p from x_0 to x). The exponential never vanishes, so W is either identically zero or never zero on the interval — a clean all-or-nothing test.
Beyond detecting independence, the Wronskian is the engine of variation of parameters: the formula for a particular solution divides by W. It appears in stability analysis, in proving uniqueness theorems, and in the theory of special-function ODEs (where Abel's identity relates the two independent solutions of, say, Bessel's equation). Named for the Polish thinker Josef Hoene-Wronski, it is one of the small set of determinants every applied mathematician carries around.
For y_1 = cos x, y_2 = sin x: W = cos x (cos x) - sin x (-sin x) = cos^2 x + sin^2 x = 1, which is never zero, confirming they are independent and form a basis for y'' + y = 0.
A Wronskian that is nonzero at even one point of the interval certifies linear independence.
For functions that are NOT solutions of a common linear ODE, a Wronskian identically zero does not imply dependence — there are textbook counterexamples; the clean dichotomy holds only for solutions of the same equation.