Solving a First Order Linear ODE System with a Constraint

[mykat]

Hello all,
I don't have much experience with ODEs.

I have a simple system, which I believe is first order linear, similar to the following:

dA/dt = 2A + 3B - C

dB/dt = A + 2B - C

dC/dt = -2A + 5B - 2C

Now I would like to include the constraint that A + B + C = 1. How do I do this mathematically?

 

[tiny-tim]

welcome to pf!

hello mykat! welcome to pf!

Now I would like to include the constraint that A + B + C = 1. How do I do this mathematically?

dA/dt + dB/dt + dC/dt = 0

 

[hunt_mat]

Or, just write C=1-A-B and insert it in the first two equations to obtain:
dA/dt=3A+2B-1
dB/dt=2A+3B-1

 

[mykat]

Thank you for the replies. I appreciate the input, I had thought to use a similar method but I wasn't sure if it was applicable.

Unfortunately, I have 7 equations and 7 variables, and as I am working with matlab, I need to have them each in the form similar to dA/dt = 3A + 4B...

Is there a more general analytical approach, rather than algebraically working out all of the equations by hand?

 

[hunt_mat]

Exponentials of matrices, so if you write in your example $\mathbf{X}=(A,B,C)^{T}$, the, you can write your equations in the form:
$$\frac{d\mathbf{X}}{dt}=\mathbf{J}\mathbf{X}$$
From here you can diagonalise your J and then solve it very easily. Can can be automatically done in matlab.

 

[mykat]

Exponentials of matrices, so if you write in your example X=(A,B,C)T, the, you can write your equations in the form:
dXdt=JX

From here you can diagonalise your J and then solve it very easily. Can can be automatically done in matlab.

By T do you mean transpose? If so, I initially had the matrices in that form. After that I wanted to add the A + B + C = 1 condition, without working out and modifying each line by hand. Is there a way to do this?

Sorry if I've completely misunderstood you.

 

[hunt_mat]

T does mean transpose. As for the A+B+C=1 condition, it's only 7 equations, or do you mean to increase it later?

 

[mykat]

Only 7 equations.

 

[hunt_mat]

Then it's not that bad then, once you've done that little hardship then you can apply my method as a quick way of solving the system.

 

[mykat]

As it turns out, the constraint was completely unnecessary. The time I wasted on this problem yesterday reflects my poor understanding of differential equations.

I am actually working with a Markov model, where the initial conditions dictate that state 1 has probability = 1 and all others are zero. Based on the nature of differential equations, probability is conserved when the system is modeled correctly.

Initially I had made a small error in the model, which gave me strange results and the false idea that I had to include a constraint. This was a great learning experience. I only wish the class I took on diff eq 2 years ago were this useful to me.

Thanks for the help anyway.

 

[hunt_mat]

So you can solve the system without any problems now?

 

[mykat]

Sure. Solving it was never the issue, it was including the unity condition, which as it happens is not necessary.