Fundamental Sets and Matrices of a Linear Homo. nth Order ODE

Fundamental Sets and Matrices of a Linear Homogeneous nth Order ODE

Consider a linear homogeneous $n^{\mathrm{th}}$ order ODE:

(1)
\begin{align} \quad y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + a_1(t)y' + a_0(t)y = 0 \end{align}

We can define what it means for a collection of solutions to this ODE to be a fundamental set of solutions.

 Definition: A Fundamental Set of Solutions to the linear homogeneous $n^{\mathrm{th}}$ order ODE $y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + a_1(t)y' + a_0(t)y = 0$ is a set of $n$ solutions $\psi_1$, $\psi_2$, …, $\psi_n$ that are linearly independent.

We can also define a corresponding fundamental matrix for this ODE.

 Definition: A Fundamental Matrix to the linear homogeneous $n^{\mathrm{th}}$ order ODE $y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + a_1(t)y' + a_0(t)y = 0$ corresponding to the fundamental set of solutions $\{ \psi_1, \psi_2, ..., \psi_n \}$ is the matrix $\Psi(t) = \begin{bmatrix} \psi_1(t) & \psi_2(t) & \cdots & \psi_n(t) \\ \psi_1'(t) & \psi_2'(t) & \cdots & \psi_n'(t) \\ \vdots & \vdots & \ddots & \vdots \\ \psi_1^{(n-1)}(t) & \psi_2^{(n-1)}(t) & \cdots & \psi_n^{(n-1)}(t) \end{bmatrix}$.

For example, consider the following second order linear homogeneous ODE:

(2)
\begin{align} \quad y'' - y = 0 \end{align}

Let $y = e^{mt}$. Then we have that:

(3)
\begin{align} \quad m^2 e^{mt} - e^{mt} &= 0 \\ \quad (m^2 - 1)e^{mt} &= 0 \end{align}

So $m^2 - 1 = 0$ which implies that $m = \pm 1$. So $\psi_1(t) = e^t$ and $\psi_2(t) = e^{-t}$ are solutions to this system. Furthermore, $\{ e^t, e^{-t} \}$ is a fundamental set of solutions to this ODE, and the corresponding fundamental matrix is:

(4)
\begin{align} \quad \Psi(t) = \begin{bmatrix} e^t & e^{-t} \\ e^t & -e^{-t} \end{bmatrix} \end{align}