Fund. Sets of Sol. to a Lin. Homo. Sys. of First Order ODEs

Fundamental Sets of Solutions to a Linear Homogeneous System of First Order ODEs

 Definition: A Fundamental Set of Solutions to the linear homogeneous system of first order ODEs $\mathbf{x}' = A(t) \mathbf{x}$ on $J = (a, b)$ is a set $\{ \phi^{[1]}, \phi^{[2]}, ..., \phi^{[n]} \}$ of linearly independent solutions to this system on $J$.

For example, consider the following linear homogeneous system of $2$ first order ODEs:

(1)
\begin{align} \quad \left\{\begin{matrix} x_1' = x_1\\ x_2' = 2x_2 \end{matrix}\right. \end{align}

We can easily solve this system. For the first differential equation:

(2)
\begin{align} \quad \frac{dx_1}{dt} &= x_1 \\ \frac{dx_1}{x_1} &= dt \\ \int \frac{1}{x_1} \: dx &= \int \: dt \\ \ln (x_1) &= t + C \\ x_1 &= e^{t + C} \\ x_1 &= C_1e^t \end{align}

Where $C_1 = e^C > 0$.

For the second differential equation:

(3)
\begin{align} \quad \frac{dx_2}{dt} &= 2x_2 \\ \frac{dx_2}{x_2} &= 2 \: dt \\ \int \frac{1}{x_2} \: dx_2 &= \int 2 \: dt \\ \ln (x_2) &= 2t + C \\ x_2 &= e^{2t + C} \\ x_2 &= C_2e^{2t} \end{align}

Where $C_2 = e^C > 0$.

Note that in fact $C_1, C_2$ can be any real numbers are we are not simply restricted to $C_1, C_2 > 0$.

Now by taking $C_1 = 1$, $C_2 = 0$ we get that $\phi^{[1]} = \begin{bmatrix} e^t\\ 0 \end{bmatrix}$ is a solution to this system. Also, by taking $C_1 = 0$ and $C_2 = 1$ we get that $\phi^{[2]} = \phi^{[1]} = \begin{bmatrix} 0\\ e^{2t} \end{bmatrix}$ is a solution to this system.

We will now show that $\{ \phi^{[1]}, \phi^{[2]} \}$ is a Fundamental set of solutions to this system on all of $\mathbb{R}$. Let $\alpha, beta \in \mathbb{R}$ and consider the following equation:

(4)
\begin{align} \alpha \phi^{[1]} + \beta \phi^{[2]} &= 0 \\ \alpha \begin{bmatrix} e^t\\ 0 \end{bmatrix} + \beta \begin{bmatrix} 0\\ e^{2t} \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ \begin{bmatrix} \alpha e^t \\ \beta e^{2t} \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{align}

The equation above implies that $\alpha e^t = 0$ and $\beta e^{2t} = 0$ for all $t \in \mathbb{R}$. Since $e^t, e^{2t} > 0$ for all $t \in \mathbb{R}$ this implies that $\alpha, \beta = 0$. So $\{ \phi^{[1]}, \phi^{[2]} \}$ is a linearly independent set of solutions to this system and so $\{ \phi^{[1]}, \phi^{[2]} \}$ is a fundamental set of solutions to this system on $\mathbb{R}$.