introduce a randomized element in a simulation
base simulation is of F = -kx, with heat introducing a random element
it's vague to me what you are simulating. i see frequency and unsure what it is the frequency of (thus, how heat could physically effect it). is it a spring? is random heat lengthening/softening the spring but random cooling also occurs? (but that situation doesn't make sense to simulate)
but if the randomness were "average random", then overall there would be no change in derivatives or integrals
without knowing what you are trying to simulate suggesting answers doesn't make sense to me
since both eq. above are solved in many books, there would be no need to solve them (you could look them up). but mm can do both forms
I'm sure Neil Singer (1st responder) knows better than me as to answering.
I see your equation as form (t->x, using y,x as typical, and Q(x) representing the random (equation of in x if we call it that, and what kind matters)):
y' + c y = Q(x)
that's certainly solvable if entered into DSolve correctly. and you can solve for Yp for each Q(x) or for fun entertain a family of Q(x) as Yp.
however for y = -kx, the "standard" eq. (time, motion, position) to use is:
y'' + w0^2 y = F sin(wt + B) (* forced undamped motion is the example, w is a constant *)
which is stolen from p. 365 of Ord. Diff. Eq. (tenenbaum, pollard). you might then wonder at the solutions of that if B is random. but it would be best to not do that and speculate for that particular case that it would result in random phase change
(if you didn't know, Y(x)=Yc+Yp, where Yp =0 if Q(x)=0, that is solutions for Q(x) are simply added to the general solution Yc which is when Q(x)=0)
MAYBE this will help answer the question. for y''+w0^2y=F sin(wt+B), if w0 is close to w, wild surges occur, otherwise we may see damping (a ring, peters out), or sin some situations a gradual rise or fall in a range (not a full complete sine wave of motion).
SUMMARY of MAYBE: if you take the opposite of what Neil Singer said (force the randomness to have a frequency) and do NOT force a frequency, then you can say nothing of the result. it could be zero by chance, increase or decrease to zero, it could decrease to zero then revive itself.