A System of Three ODEs

hzzhangyu

1. The problem statement, all variables and given/known data
In a problem I was given a system of three differential equations concerning three functions, x(t), y(t) and z(t):

dx(t)=2y(t)dt,
dy(t)=[z(t)-x(t)]dt,
dz(t)=[c^2x(t)-2y(t)]dt. (where c is a constant)

The problem asked me to prove that when t is large, x(t)+z(t) converges to K*exp{wt},
where w is a positive real root of equation w^3+4w-2c^2=0,

2. Relevant equations



3. The attempt at a solution

I haven't studied how to solve this kind of ODE system in my calcus class, so now I am stuck at the beginning of this question. If you are willing to take time to look at it for me, I will be real grateful for that. Thanks!

vela

The middle equation is

y'(t) = z(t)-x(t).

If you differentiate it, you'll get

y''(t) = z'(t)-x'(t)

You can substitute for x'(t) and z'(t) using the first and third equations. With a bit more manipulation, you can eventually get a differential equation for just y(t), which you should be able to solve (in principle). Next, write (x+z)'' in terms of y. Then you should be able to argue what you're trying to show.