How to solve a linear differential equation with a random coefficient in Mathematica

Question

I have a differential system like

dx/dt = A x(t) + B y(t)

dy/dt = C x(t) + D y(t)

where A, B, C, and D are real constants. Now I need to explore the behavior of the system if A, instead of being a constant number, is a random number uniformly distributed between a given range. I just need to check qualitatively. I have no background on stochastic integrals, therefore I do not know if this is actually something related with the Ito integral (and this question https://mathematica.stackexchange.com/questions/3141/how-can-you-compute-it-integrals-with-mathematica) . In any case, I do not know how to solve this differential equation.

Any guidance is highly appreciated.

Answer 1

The standard way to solve your system is

FullSimplify[ 
        DSolve[{y'[t] == a x[t] + b y[t], x'[t] == c x[t] + d y[t]}, {y, x}, t]]

Now, you should think WHAT do you want to explore when {a, b, c, d} are random parameters.

Edit

Perhaps you want something like this:

s = FullSimplify[
     DSolve[{y'[t] == #[[1]] x[t] + #[[2]] y[t], x'[t] == #[[3]] x[t] + #[[4]] y[t], 
            x[0] == 1, y[0] == 1}, {y, x}, t]] & /@ RandomReal[{-1, 1}, {30, 4}];

ParametricPlot[Evaluate[{x[t], y[t]} /. s[[All, 1]]], {t, 0, 1}]