# Thread: control systems - ODE confusion

1. ## control systems - ODE confusion

Suppose that is the solution to the initial value problem

where is the state transition matrix.

Find for the following

I've done:

So

But since the left side is with respect to x_1 and not x_2 I'm not sure I can do this, and not sure where to go from here?

2. You have x1'= -x1- 4x2 and x2'= -x1- x2. From the first equation, 4x2= -x1'- x1. If you differentiate the first equation again, you get x1"= -x1'- 4x2'. From the second equation, x2'= -x1- x2 so that is x1"= -x1'- 4(-x1- x2)= -x1'- 4x1+ 4x2= -x1'- 4x1+ (-x1'- x1)= -2x1'- 5x1.

Can you solve the single equation x1"= -2x1'- 5x1 which is the same as x1"+ 2x1'+ 5x1= 0? If so then you can find x2 from 4x2= -x1'- 1.

Another way to do this problem, more in the "matrix" spirit, would be to find the eigenvalues and eigenvectors of $\begin{bmatrix}-1 & -4 \\ -1 & -1\end{bmatrix}$

3. Thank you so much that post was really helpful.

I'm having trouble with the same question but for matrix:

I've done:

So

which gives rise to:

However I can't see anyway of finding out what x_3 is, am I missing something?!

Thanks again.