592 CHAPTER 30. LAPLACE TRANSFORM METHODS
Now I compute this.
X (s) =
− 1(s2+1)(s−1)3
(−s4 + s3 +2
)1
(s−1)2s2−3s+4
s3−2s2+s−21
(s−1)3(s3−2s2+s−2)
(s5−3s4 +3s3−2s2 + s+2
)
At this point, I use partial fractions and go backwards in the table or I ask a computeralgebra system to find the inverse Laplace transform. I recommend using the computeralgebra system. Thus
x(t) =
12 cos t + 1
2 et − 12 t2et + 3
2 tet
12 et − 9
10 cos t− 310 sin t + 2
5 e2t − tet
15 cos t− 1
10 sin t + 45 e2t − 1
2 t2et + 12 tet
This is then the solution to the first order system. I used Scientific Notebook to do all ofthese computations. However, one can also use Matlab. You will need Matlab and thesymbolic math toolbox installed for this to work.
>>syms s t; a=(enter initial vector here); b=(enter sI-A 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=[s-2 -2 1;1 s -1;1 0 s-2]; 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 thestart 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))/2The 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 whichis part of the symbolic math toolbox in Matlab.
Example 30.2.9 Find the fundamental matrix of
A =
−3 2 −10 −1 14 −4 3
and use to solve the initial value problem
x′ = Ax+
ln(t2 +1
)sin(t2)
cos(t)
, x(0) =
111
It will not be possible to give a closed form solution for this problem but we can write
it in terms of an integral if the fundamental matrix is found.