# Exp Matrix to solve an initial value problem

1. Sep 23, 2008

### Alphaboy2001

Exp Matrix to solve an initial value problem(urgent)

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

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)

3. 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

Last edited: Sep 23, 2008