Analytical Solution to this? - linear system of ODES

1. Oct 7, 2008

Analytical Solution to this? -- linear system of ODES

Hi All,

It's been awhile since I've even attempted to solve something analytically, so before jumping back into the text. Does the following already have a common solution that I can find somewhere?

Thanks,

dx1/dt = A1 + B1x1 +C1x2 + D1x3
dx2/dt = A2 + B2x1 +C2x2 + D2x3
dx3/dt = A3 + B3x1 +C3x2 + D3x3

2. Oct 7, 2008

HallsofIvy

Staff Emeritus
Re: Analytical Solution to this? -- linear system of ODES

That certainly does have an "analytical solution" since it is a system of linear we equations with constant coefficients. If you are asking for a specific formula, so yo can just plug in the values of A1, B1, etc. I doubt you will find such a thing. I would be inclined to write it as a matrix equation:
$$\frac{d\left[\begin{array}{c}x1 \\ x2 \\ x3 \end{array}\right]}{dt}= \left[\begin{array}{ccc} B1 & C1 & D1 \\ B2 & C2 & D2 \\ B3 & C3 & D3\end{array}\right]\left[\begin{array}{c} x1 \\ x2 \\ x3 \end{array}\right]+ \left[\begin{array}{c}A1 \\ A2 \\ A3\end{array}\right]$$
and then look for the eigenvalues of the matrix. The eigenvalue equation is, of course, a cubic equation.

3. Oct 7, 2008

Re: Analytical Solution to this? -- linear system of ODES

I assumed since it was linear that I could get a solution to it, but like you said there is a cubic in there. That cubic is probably going to be a pain in the ...
I was especially asking, because for whatever reason it looks like it could be a really standard problem that has been solved over and over again. Before diving in, I just wanted to test the waters.

Also, why did you quote "analytical solution"? Just curious...

Thank you