|
Example 1 Solve
the following system.
Solution
First notice that the system is not given in matrix
form. This is because the system won’t
be solved in matrix form. Also note
that the system is nonhomogeneous.
We start just as we did when we used Laplace
transforms to solve single differential equations. We take the transform of both differential
equations.
Now plug in the initial condition and simplify things a
little.
Now we need to solve this for one of the transforms. We’ll do this by multiplying the top
equation by s and the bottom by -3
and then adding. This gives,
Solving for X1
gives,
Partial fractioning gives,
Taking the inverse transform gives us the first solution,
Now to find the second solution we could go back up and
eliminate X1 to find the
transform for X2 and
sometimes we would need to do that.
However, in this case notice that the second differential equation is,
So, plugging the first solution in and integrating gives,
Now, reapplying the second initial condition to get the
constant of integration gives
The second solution is then,
So, putting all this together gives the solution to the
system as,
|