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Consider the linear ODE system:

$\dot x = \begin{pmatrix} 1 & -1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}x, x(t_0)=x_0 $

(i) Calculate the general solution

(ii) Calculate the particular solution when $x(0)=(0,0,1)^T$

(iii) Calculate the restriction of the dynamics to the unstable subspace $E^u$.

I believe I have the general solution for (i) with

$e^{tA}=e^t\begin{pmatrix} 1 & -t & t-t^2 \\ 0 & 1 & 2t \\ 0 & 0 & 1 \end{pmatrix}$

(my working is quite long but I could add it if necessary)

so the general solution would be

$x(t)= e^t\begin{pmatrix} 1 & -t & t-t^2 \\ 0 & 1 & 2t \\ 0 & 0 & 1 \end{pmatrix}x(0)$?

for (ii) I replaced x(0) with the conditions to get:

$x(t)= e^t\begin{pmatrix} 1 & -t & t-t^2 \\ 0 & 1 & 2t \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}=e^t\begin{pmatrix} t-t^2 \\ 2t \\ 1 \end{pmatrix}$?

but I'm really not sure where to go from here, any help?

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  • $\begingroup$ You say that your working to find the general solution is long. If you decompose the coefficient matrix into the sum of the identity and a nilpotent matrix, you can compute its exponential directly and fairly easily, without having to find any generalized eigenvectors. $\endgroup$
    – amd
    Jun 17, 2018 at 17:59
  • $\begingroup$ What exactly is your question? You’ve already computed both the general and particular solutions, contrary to the title of the question. $\endgroup$
    – amd
    Jun 17, 2018 at 18:01

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