I'm not sure how to find the first derivative of the Wronskian. I have the equation of the Wronskian for two functions where I only use the functions and their first derivatives.
I have the following:
$$\underline{\overline{X}}(t) = [x^{(1)}(t), x^{(2)}(t)]$$
is the solution to
$$\frac{d\underline{\overline{X}}}{t} = A(t)\underline{\overline{X}}(t)$$
where
\begin{equation*} A(t) = \begin{pmatrix} a_{11}(t) & a_{12}(t) \\ a_{21}(t) & a_{22}(t) \end {pmatrix} \end{equation*}
I calculated the Wronskian by taking the determinant of the derivative matrix of $\underline{\overline{X}}(t)$ and got: $$x^{1}(x^{(2)})' - x^{2}(x^{(1)})'$$
We were told in class that the first derivative of the Wronskian will always be
$$\text{trace}(A)W$$
but I'm not sure how to get this result from differentiating what I got for the Wronskian in this case.
Any help is appreciated!
Given the differential equation
$\dot X = AX, \tag 0$
where
$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \tag 1$
and
$X = \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix}, \tag 2$
we have
$\dot X = \begin{bmatrix} \dot x_{11} & \dot x_{12} \\ \dot x_{21} & \dot x_{22} \end{bmatrix} = AX = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}\begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix} = \begin{bmatrix} a_{11}x_{11} + a_{12}x_{21} & a_{11}x_{12} + a_{12}x_{22} \\ a_{21}x_{11} + a_{22}x_{21} & a_{21}x_{12} + a_{22}x_{22} \end{bmatrix}; \tag 3$
we define the Wronskian
$W = \det \left ( \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix} \right ) = x_{11}x_{22} - x_{12}x_{21}; \tag 4$
we compute, using (3):
$\dot W = \dot x_{11}x_{22} + x_{11} \dot x_{22} - \dot x_{12}x_{21} - x_{12} \dot x_{21}$ $= (a_{11}x_{11} + a_{12}x_{21})x_{22} + x_{11}(a_{21}x_{12} + a_{22}x_{22}) - (a_{11}x_{12} + a_{12}x_{22})x_{21} - x_{12}(a_{21}x_{11} + a_{22}x_{21})$ $= a_{11}x_{11}x_{22} + a_{22}x_{11}x_{22} - a_{11}x_{12}x_{21} - a_{22}x_{12}x_{21}$ $= a_{11}(x_{11}x_{22} - x_{12}x_{21}) + a_{22}(x_{11}x_{22} - x_{12}x_{21}) = (a_{11} + a_{22})(x_{11}x_{22} - x_{12}x_{21}) = \text{Tr}(A) W. \tag 5$