TITLE: What is the intuitive meaning of the Wronskian?
QUESTION [11 upvotes]: What is the intuition behind the Wronskian?  I do not want an explanation that involves a rigorous proof. I just want to learn how to intuitively think about it.
Edit: And since it is tied to the weight function in the theory of differential equations, it would also be great to have an intuitive connection between the two included in the answer!

REPLY [5 votes]: To me, the key formula in understanding the Wronskian of a linear homogeneous first order differential system is the following: 
$$
\text{if}\quad \frac{d A}{dt}(t)=B(t)A(t)\quad \text{then}\quad \frac{d}{dt}\left(\det A(t)\right) = \mathrm{tr}\, B(t)\det A(t).$$ 
Here $A, B$ are square matrices and $\mathrm{tr}$ stands for "trace" and this formula is often called "Liouville's formula".
You apply it as follows. Consider the system of $n$ differential equations 
$$
\frac{d\boldsymbol{x}}{dt} = B(t) \boldsymbol x(t).$$
(Boldface denotes elements of $\mathbb R^n$, considered as column vectors). Taking $n$ solutions $\boldsymbol{x}_1\ldots \boldsymbol{x}_n$ one forms the matrix 
$$
A(t)=\begin{bmatrix} \boldsymbol{x}_1,\ \ldots\, , \boldsymbol{x}_n\end{bmatrix}\in\mathbb{R}^{n\times n}.$$

The Wronskian determinant $\det A(t)$ is geometrically interpreted as the oriented volume of the parallelepiped spanned by $\boldsymbol{x}_1\ldots \boldsymbol{x}_n$. Liouville's formula says that this volume's evolution is governed by the trace of $B(t)$.

For a single differential equation of order $n$ 
$$
x^{(n)}(t)+ b_{n-1}(t)x^{(n-1)}(t)+\ldots +b_0(t) x(t) = 0$$ 
one introduces the vector 
$$
\boldsymbol{x}(t)= \begin{bmatrix} x(t) \\ x'(t) \\ \vdots \\ x^{(n-1)}(t)\end{bmatrix}$$ 
and the matrix ("companion matrix") 
$$
B(t)=\begin{bmatrix} 0& 1& 0& \ldots& \ldots& \ldots \\ 
0 & 0 & 1 & 0 & \ldots&\ldots \\ 
& &  &\vdots &  & \\
0& \ldots& \ldots& \ldots& 0 & 1 \\ -b_0(t) & -b_1(t)& \ldots &\ldots & -b_{n-2}(t) & -b_{n-1}(t)\end{bmatrix}$$
so that the equation is equivalent to $\frac{d\boldsymbol{x}}{dt}=B(t)\boldsymbol{x}(t)$. Often, one says that $\boldsymbol{x}$ belongs to the phase space of the original equation. (This is especially true when $n=2$, in which case the phase space is typically interpreted the Cartesian product of position and velocity.) 

In the case of a linear homogeneous differential equation of order $n$, the Wronskian is interpreted as an oriented volume in phase space. Liouville's formula says that the evolution of the Wronskian is governed by the term $\mathrm{tr}\, (B(t))=-b_{n-1}(t)$. 

Further development. This interpretation fits well in the frame of Liouville's theorem for Hamiltonian systems. See also here.