You want to solve

Use the Property 21. and take Laplace transforms of both sides. Thus

where X

Then

Note that there is even a formula for
Example 24.3.1 Solve the following first order system.

Following the above general procedure, the Laplace transform of the forcing function is

and so

Now I compute this.

At this point, I use partial fractions and go backwards in the table or I ask a computer algebra system to find the inverse Laplace transform. I recommend using the computer algebra system. Thus

This is then the solution to the first order system. I used Scientific Notebook to do all of these computations. When I found the Laplace transform of the forcing function, I put the cursor next to the forcing function and clicked on compute then transforms and then Laplace. To go backwards and find the inverse Laplace transform, I did the same thing except at the end I clicked on inverse Laplace. If it had not been possible to find the eigenvalues exactly, this would not have worked out because the computer algebra system could not find what it needed. Actually, Matlab can do it all numerically including finding the inverse Laplace transform, but it becomes a little ugly since it uses a method from complex analysis to numerically find the inverse Laplace transform. When this happens, you are likely better off simply having Matlab find the solution numerically. I will discuss this later. First, here is how you use Matlab. You will need Matlab and the symbolic math toolbox installed for this to work.
>>syms s t; a=(enter initial vector here); b=(enter sIA here); c=(enter f(t) here);
simplify(ilaplace(inv(b)*(a+laplace(c))))
I will use this to solve the above problem.
>> syms s t; a=[1;0;1]; b=[s2 2 1;1 s 1;1 0 s2]; c=[cos(t);sin(t);exp(t)];
simplify(ilaplace(inv(b)*(a+laplace(c))))
Note the use of square brackets in entering the matrix. You must use these. You enter one row at a time with a space between successive entries and a semicolon to indicate the start of a new row. Then you press enter on your keyboard and it will produce the following:
cos(t)/2 + exp(t)/2  (tˆ2*exp(t))/2 + (3*t*exp(t))/2
(2*exp(2*t))/5  (9*cos(t))/10 + exp(t)/2  (3*sin(t))/10  t*exp(t)
(4*exp(2*t))/5 + cos(t)/5  sin(t)/10  (tˆ2*exp(t))/2 + (t*exp(t))/2
The advantage to using Scientific notebook is the result comes out looking a lot nicer but you get the same thing either way. In fact Scientific notebook is based on mupad which is part of the symbolic math toolbox in Matlab.