Matrix equation, solving for x(t)

[exidez]

this comes strait from a textbook:
http://higheredbcs.wiley.com/legacy/college/nise/0471794759/appendices/app_i.pdf
I am looking at how they obtained (I.24) from (I.25) on page 3 and 4.

Firstly we have:

$e^{-\textbf{A}t}x(t)-x(0)=\int{e^{-\textbf{A}t}\textbf{Bu}(\tau)d\tau}$

Then this is derived to:

$x(t)=e^{-\textbf{A}t}x(0)+\int{e^{-\textbf{A}(t-\tau)}\textbf{Bu}(\tau)d\tau}$

my question is why does the derived equation have $e^{-\textbf{A}t}$. I thought it would have been $e^{\textbf{A}t}$ instead. Can someone explain this too me. The text didnt help me.

 

[AlephZero]

I think there are typos in the first line of equation 25. Some of the minus signs shouldn't be there.

The statement "where $\Phi(t) = e^{At}$ by defintion" and the equation involving $\Phi$ are correct.

Just take equation 24 and multiply all the term by $e^{At}$ (note, no minus sign!) to get the correct version.