System of differential equations
Hi guys, suppose we have the following system of differential equations
X' = -aX + cY
Y' = aX - (b+c)Y + dZ
Z' = bY - dZ
where a,b,c,d > 0 and constant.
We can show that the system has non-zero constant solutions and that these satisfy
X/Y = c/a , Y/Z = d/b.
(This is easy to show)
But what I now need to do is show that for any initial data, the solution will tend to one of these solutions as t approaches infinity.
I have tried to find the eigenvalues for the system(using MAPLE) but this gets messy fast.
If anyone could help it would be much appreciated, cheers.
Re: System of differential equations
Note that X'+Y'+Z'=0 hence X+Y+Z=C
Bringing back Y=C-X-Z into the first and tbe third equation reduces to a system of two equations where the unknown are X and Z, easier to solve. Various methods are allowed (Matricial, or Laplace transform, or substitution)
For example, sustitution leads to
Solving this ODE leads to a general solution on the form :
X(t) = Xoo+C1*exp(k1*t)+C2*exp(k2*t)
Xoo is a constant term.
The coeffients C1 and C2 depend on the initial condition, but it doesn't matter.
k1 and k2 are rather simple functions of a, b, c, d which can be easily expressed.
It is then possible to show that k1 and k2 are negative as far as a, b, c, d are positive. So, when t tends to infinity, the exponential terms tends to 0. Only the constant term remains, which is the limit as t approches the infinity.