Linear Homo. and Linear Nonhomo. Systems of First Order ODEs

Linear Homogeneous and Linear Nonhomogeneous Systems of First Order ODEs

Linear Homogeneous Systems of First Order ODEs

Definition: Let $J = (a, b)$, $A : J \to \mathbb{R}^{n \times n}$ be a continuous function on $J$. A Linear Homogeneous System of $n$ First Order Ordinary Differential Equations on the interval $J$ is of the form $\mathbf{x}' = A(t) \mathbf{x}$ where $\mathbf{x}' = \begin{bmatrix} x_1'\\ x_2'\\ \vdots \\ x_n' \end{bmatrix}$, $A(t) = \begin{bmatrix} a_{1,1}(t) & a_{1,2}(t) & \cdots & a_{1,n}(t) \\ a_{2,1}(t) & a_{2,2}(t) & \cdots & a_{2,n}(t)\\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1}(t) & a_{n,2}(t) & \cdots & a_{n,n}(t) \end{bmatrix}$, and $\mathbf{x} = \begin{bmatrix} x_1\\ x_2\\ \vdots \\ x_n \end{bmatrix}$. The system above is said to also be Autonomous if $A$ is an $n \times n$ matrix of constant functions on $J$.

We will use the boldface notation in representing the system $\mathbf{x}' = A(t)\mathbf{x}$. We will not use boldface notation to represent the solutions of such systems though.

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

(1)

\begin{align} \quad \left\{\begin{matrix} x_1' = 2tx_1 + t^2x_2\\ x_2' = t^3x_1 + 4tx_2 \end{matrix}\right. \end{align}

This is a linear homogeneous system of $2$ first order ODEs with $A(t) = \begin{bmatrix} 2t & t^2\\ t^3 & 4t \end{bmatrix}$.

Linear Nonhomogeneous Systems of First Order ODEs

Definition: Let $J = (a, b)$, $A : J \to \mathbb{R}^{n \times n}$ be a continuous function on $J$ and $g : J \to \mathbb{R}^n$ be a continuous function on $J$ where $g$ is not identically zero on $J$. A Linear Nonhomogeneous System of $n$ First Order Ordinary Differential Equations on the interval $J$ is of the form $\mathbf{x}' = A(t) \mathbf{x} + g(t)$ where $\mathbf{x}' = \begin{bmatrix} x_1'\\ x_2'\\ \vdots \\ x_n' \end{bmatrix}$, $A(t) = \begin{bmatrix} a_{1,1}(t) & a_{1,2}(t) & \cdots & a_{1,n}(t) \\ a_{2,1}(t) & a_{2,2}(t) & \cdots & a_{2,n}(t)\\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1}(t) & a_{n,2}(t) & \cdots & a_{n,n}(t) \end{bmatrix}$, $\mathbf{x} = \begin{bmatrix} x_1\\ x_2\\ \vdots \\ x_n \end{bmatrix}$, and $g(t) = \begin{bmatrix} g_1(t)\\ g_2(t)\\ \vdots\\ g_n(t) \end{bmatrix}$.

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

(2)

\begin{align} \quad \left\{\begin{matrix} x_1' = 2tx_1 + t^2x_2 + \sin t\\ x_2' = t^3x_1 + 4tx_2 + e^t\cos t \end{matrix}\right. \end{align}

This is a linear nonhomogeneous system of $2$ first order ODEs with $A(t) = \begin{bmatrix} 2t & t^2\\ t^3 & 4t \end{bmatrix}$ and $g(t) = \begin{bmatrix} \sin t\\ e^t \cos t\end{bmatrix}$.