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

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)

For example, sustitution leads to

X''+(a+b+c+d)X'+(ab+ad+cd)X=Ccd

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.