Fundamental Matrices in Differential Equations

[rocketboy]

The question:

Find the fundamental matrix for the given system of equations, and then find the specific solution to satisfy x(0) = I.

x' = \left(\stackrel{1}{5}\stackrel{ -1}{ -3}\right)x

My solution:

I found the eigenvalues to be r_{1}= -1 + i and r_{2}= -1 - i

I then found the corresponding eigenvectors, and obtained a general solution as follows:

x = c_{1}e^{-t}\left(\stackrel{cos(t)}{2cos(t)+sin(t)}\right) + c_{2}e^{-t}\left(\stackrel{sin(t)}{2sin(t)-cos(t)}\right)

I know that this is correct, as I have checked it with a solution manual. However the solution manual becomes difficult to comprehend past this point as it skips many steps.

I now have to find the specific solution for x(0) = I.

I set the above equal to \left(\stackrel{1}{0}\right)(cos(t) + i sin(t)) in order to solve for the first specific solution, however can't seem to solve it, and I have no idea what the solution manual is doing. They come up with the following as a specific solution:

x = e^{-t}\left(\stackrel{cos(t) + 2sin(t)}{5sin(t)}\right)

and then proceed to find the specific solution for the second part of the identity matrix, and finally the fundamental solution is the matrix involving these two specific solutions.

The part I don't understand is how they got the specific solution above. Any help would be greatly appreciated!

Thank-you,
-J

 

[rocketboy]

Alright so apparently doing math at 3am was the problem, not my understanding. I woke up this morning and realized when I was attempting to solve the initial value problem I wasn't substituting '0' for 't'... when I did it all made sense and took a couple minutes to come up with the answers they did.

Thanks anyway!