# System of ODEs

1. Nov 18, 2013

### jimmycricket

1. The problem statement, all variables and given/known data

Given the matrix $b=\begin{pmatrix}-1&0&-1\\-4&3&-1\\0&0&-2\end{pmatrix}$ decide if the system of ODEs, $\frac{dx}{dt}=Bx$ is decoupled. If yes find the general solution x=xh(t)

2. Relevant equations

3. The attempt at a solution
I would say the matrix is decoupled since the second equation involving $2x$2(t) can be solved without the other two equations. Then the third equation can be solved without knowing $x$1(t). We have:

x1(t)-x2(t)-3x3(t)=x'1
2x2(t)=x'2
x2(t)+4x3(t)=x'3

Im not sure where to go from here.

2. Nov 18, 2013

### pasmith

How did you get that? You have
$$\begin{pmatrix} \dot x_1 \\ \dot x_2 \\ \dot x_3 \end{pmatrix} = \begin{pmatrix}-1&0&-1\\-4&3&-1\\0&0&-2\end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$$
Hence
$$\dot x_1 = -x_1 - x_3 \\ \dot x_2 = -4x_1 + 3x_2 - x_3 \\ \dot x_3 = -2x_3$$

3. Nov 19, 2013

### jimmycricket

yes thats what I wrote down on paper. There was a bit of a mistranslation when trying to write it in latex.