Solving Decoupled System of ODEs with Matrix b

[jimmycricket]

Homework Statement

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=x_h(t)

Homework Equations

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.

 

[pasmith]

Homework Statement

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=x_h(t)

Homework Equations

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.

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
$$\dot x_1 = -x_1 - x_3 \\ \dot x_2 = -4x_1 + 3x_2 - x_3 \\ \dot x_3 = -2x_3$$

 

[jimmycricket]

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