We've been given this tricky question and I have a few ideas jotted down but when I've gone to try them they haven't been making sense and I've been going round in circles and it's truly bugging me now so I need somebodies help please!! The question is:
Suppose we have two species interacting on the same environment. After observations we find that the size of their populations obey the following set of equations:
dW/dt = αW−βV
dV/dt = −γV+δW
where α,β,γ,δ are positive constants therefore solve the system by proposing
W (t) = A sin(ωt) + B cos(ωt)
V (t) = C sin(ωt) + D cos(ωt)
Find the values of the constants A,B,C and D ONLY in terms of the parameters α,β,γ,δ ω
So basically I was thinking maybe it involves subbing them into the original system, differentiating them and comparing coefficients and then solving a system of equations (however this doesn't seem to work as you can only get the constants A,B,C and D in terms of a MIX of constants and parameters and we solely what parameters and plus you only obtain 4 homogenous equations implying that at least 1 of the constants must equal zero so that's got to incorrect or is it?
Now I was thinking perhaps converting them into harmonic form and then playing around it with but then I come to the same conclusion as above pretty much where I get a mix of constants and parameters when I only want parameters.
So maybe involving an idea to do with these set of differential equations representing a second order equation or some sort (A second order equation and its solutions)
but I'm just going round in circles any help please guys!