__Central differences are used for changing the parameters__:

This looks for the first Parameter (**vol1**) as follows:

Parameterchange for vol1 **left** from the central point

A simple general solution requires the exponential of a matrix. Easy enough to do with the eigenvector decomposition of a matrix. But what we really need is for a given A and many values of t. Eigenvector calculations are somewhat time-consuming. So we design a function that can, given A, create the function as a function of t.

Now we do a bit of general calculus. Consider the differential equation where A and B are constants. The solution will be of the form where C and D are constants. We can find D by differentiating the equation for y and equating it to the value from the ODE, and then equating terms. We get . We can solve that last equality for D (the y's cancel out) and get . We can find C from the inital (or boundary) value. If we have that then we can substitute and solve to get . Using these two solutions we can write a function to get C and D: