Method to solve a system of differential equations

Question

I'm studying systems of linear equations.

I'm now specifically studying systems of linear equations of the 1st order, homogeneous:

$Y' = AY$

$A$ as a constant matrix.

Now I know there are various methods to solve this systems. My professor talked about one of the method that consists on reducing the system into a single equation. Does anyone knows where can I find notes or explanations about how to apply correctly this method? I'm having trouble finding it on the internet. I find a lot of things about the method that uses eigenvalues/eigenvectors but that's a method that my professor doesn't like... Does anyone know a good text that I can look for?

Answer 1

I'm not entirely sure what your professor is hinting at, perhaps ask him/her to elaborate. I am assuming that $y$ is a vector. A homogenous differential equation of the form \begin{align} y' = Ay \end{align} has a solution of the form \begin{align} e^{At}y(0) \end{align} The procedure is similar to the one-dimensional case. Read more about the matrix exponential and how it is useful in solving DEs.

Answer 2

If there is a $b$ such that $\{b,Ab,...,A^{n-1}b\}$ is a basis (that is, if the pair $(A,b)$ is completely controllable), then we can reduce the system to a single differential equation.

In the above basis, $A$ has the form (controllable canonical form) \begin{bmatrix} 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0 \\ 0 & 0 & \cdots & 0 & 1 \\ -a_n & -a_{n-1} & \cdots & -a_2 & -a_1 \end{bmatrix} and the system of differential equations looks like $\dot{x}_1 = x_2,..., \dot{x}_{n-1} = x_n$, $\dot{x}_n = -a_n x_1 - \cdots - a_1 x_n$, and by expanding we get the equation $x_1^{(n)} + \sum_{k=0}^{n-1} a_{n-k}x_1^{(k)} = 0$, which is a $k$th order differential equation in $x_1$.

Such a $b$ exists iff in the Jordan normal form of $A$, there is exactly one Jordan block of $A$ associated with any eigenvalue.