Wronskians: Independence Tests and Abel’s Identity

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What is the Wronskian of two functions?

For differentiable functions $f$ and $g$, it is $W(f,g)=fg'-gf'$, the determinant with rows $(f,g)$ and $(f',g')$.

How is the Wronskian formed for $n$ functions?

It is the determinant whose columns correspond to the functions and whose rows contain derivatives from order $0$ through order $n-1$.

What does a nonzero Wronskian at one point establish?

It establishes that the functions are linearly independent on the interval containing that point.

What does linear dependence imply about the Wronskian?

A constant linear dependence among the functions also holds among their derivatives, so the Wronskian vanishes identically.

What is Abel’s identity for $y''=a(x)y'+b(x)y$?

Solutions have a Wronskian satisfying $W'=a(x)W$, hence $W=Ce^{A(x)}$ for $A'=a$.

Quiz questions

1. If $y_1$ and $y_2$ solve $y''=a(x)y'+b(x)y$, what equation does their Wronskian satisfy?Answer: $W'=a(x)W$.
2. For solutions of $y''+P(x)y'+Q(x)y=0$, which equation follows for the Wronskian?Answer: $W'=-P(x)W$.
3. Suppose $y_2$ is a known solution and is nonzero on an interval. Which first-order equation follows from $W=y_1y_2'-y_2y_1'$?Answer: $y_1'-\frac{y_2'}{y_2}y_1=-\frac{W}{y_2}$.
4. For solutions of the same monic $n$th-order homogeneous linear differential equation, what follows if the solutions are linearly independent on an interval?Answer: Their Wronskian is nowhere zero on that interval.
5. In a generalized Wronskian for functions of several variables, what is applied to the functions to form matrix entries?Answer: Suitable constant-coefficient linear partial differential operators.