Exp Matrix to solve an initial value problem

[Alphaboy2001]

Exp Matrix to solve an initial value problem(urgent)

Homework Statement

Given the matrix $$A = \left[\[\begin{array}{ccc} 2 & 2 \\ 1& 3 \end{array}\right]$$

a) Find the $$e^{tA}$$

2) Solve the $$x' = Ax + (1,0) \begin{array}{c} \end{array}$$ where x(0) = (0,0)

The Attempt at a Solution

a) $$e^{tA} = P_{A} \cdot e^{Dt} \cdot P'$$

Which in my book gives

$$e^{tA} = \left[\[\begin{array}{ccc} -2 & 1 \\ 1& 1 \end{array}\right] \cdot \left[\[\begin{array}{ccc} e^{t} & 0 \\ 0& e^{16t} \end{array}\right] \cdot \left[\[\begin{array}{ccc} -\frac{1}{3} & \frac{1}{3} \\ \frac{4}{3}& \frac{8}{3} \end{array}\right]$$

$$e^{tA} = \left[\[\begin{array}{ccc} \frac{2}{3}\cdot e^{t} + \frac{4}{3}\cdot e^{16t} & \frac{-2}{3}\cdot e^{t} + \frac{8}{3}\cdot e^{16t} \\ \frac{-1}{3}\cdot e^{t} + \frac{4}{3}\cdot e^{16t} & \frac{1}{3}\cdot e^{t} + \frac{8}{3}\cdot e^{16t} \end{array} \right]$$

Doesn't that look okay??

b) From what I remember the solution for x' can be written as $$X = e^{tA} \cdot C$$

Which in my case gives $$X = e^{tA} \cdot \left[\begin{array}{c} 0 \\ 0 \end{array} \right]$$

This is how my textbook argues how solve such eqn, but if I fry to x' I totally different result. What am I doing wrong? Or could somebody please be so kind to lead me on the right path/track?? :)

Sincerely
Alphaboy

 

[mighty2000]

Dear Alphaboy,

Thank you for your post. It seems like you are on the right track with your solution for part a). Your matrix for e^{tA} looks correct. However, for part b), there may be some confusion with the notation. The solution for x' = Ax + (1,0) with x(0) = (0,0) can be written as x(t) = e^{tA} \cdot x(0) + e^{tA} \cdot \int_{0}^{t} e^{-sA} \cdot (1,0) ds. In this case, x(0) = (0,0), so the first term becomes 0. The second term can be simplified to e^{tA} \cdot \int_{0}^{t} e^{-sA} ds. You can use your solution for e^{tA} from part a) to evaluate this integral. I hope this helps. Good luck with your problem!


Scientist