# System of differential equations

• Aug 10th 2012, 09:12 PM
liedora
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.
• Aug 11th 2012, 01:36 AM
JJacquelin
Re: System of differential equations
Hi !
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)