now that weve found some of the fundamentals out

Question

Now that we've found some of the fundamentals out of the way for systems of differential equations it's time to start thinking about how to solve a system of differential equations. We will begin with the homogeneous system written in matrix form as,

x?' = A x?    ......................(1)

Here, A is an n x n matrix and x is a vector whose elements are the unknown functions into the system.

Here, if we begin with n = 1 then the system decreases to a fairly easy linear or separable first order differential equation,

x' = ax

And it has the following solution,

 x′ = ax

x (t) =  ce^at

Therefore, let's use this as a guide and for a common n let's notice if,

x? (t) = ?h   e^rt    .................(2)

It will be a solution. Remember that the only real difference now is which we let the constant in front of the exponential be a vector. All we requirement to do then is plug it into the differential equation and notice what we find.  First see that the derivative is,

x? (t) = r ?he^rt   

Therefore upon plugging the guess in the differential equation we find,

r ?he^rt = A ?he^rt

(A - rI) ?he^rt =0?

Here, as we know that exponentials are not zero we can drop which portion and we after that see that so as for (2) to be a solution to (1) so we should have,

(A - rI) ?h = 0?

Or, so as for (2) to be a solution to (1), r and ?h should be an eigen-value and eigenvector for the matrix A.

Thus, so as to solve (1) we first get the eigen-values and eigenvectors of the matrix A and after that we can form solutions by using (2). There are going to be three cases which we'll require to look at.

The cases are as: real, distinct eigenvalues, complex eigenvalues and repeated eigenvalues.

None of that tells us how to wholly solve a system of differential equations. We'll require the subsequent couple of facts to do this.