📂Odinary Differential Equations

First-Order Linear Differential Equation System

Buildup¹

When the mass is $m$, the damping factor is $\gamma$, and the spring constant is $k$, the equation of motion representing the vibration of an object hung on a spring is as follows.

$$m x^{\prime \prime} + \gamma x^{\prime} + kx = F$$

Letting $x_{1}=x$, $x_{2}=x_{1}^{\prime}$, the above equation of motion can be expressed as the following system.

$$\begin{align*} x_{1}^{\prime}(t) =&\ x_{2}(t) \\ x_{2}^{\prime} (t) =&\ x_{1}^{\prime \prime}(t) = -\dfrac{\gamma}{m}x_{2}(t)-\dfrac{k}{m}x_{1}(t)-\dfrac{1}{m}F(t) \end{align*}$$

This can be represented as a matrix as follows.

$$\begin{align*} && \begin{pmatrix}x_{1}^{\prime} \\ x_{2}^{\prime}\end{pmatrix} =&\ \begin{pmatrix} 0 & 1 \\ -\dfrac{k}{m} & -\dfrac{\gamma}{m} \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} + \begin{pmatrix} 0 \\ -\dfrac{1}{m}F \end{pmatrix} \\ \implies&& \mathbf{x}^{\prime}(t) =&\ A\mathbf{x}(t)+g(t) \end{align*}$$

In the case of $g(t)=0$ being a homogeneous equation, it can be seen that solving a second-order differential equation simplifies to solving a matrix multiplication problem with solution $\mathbf{x}^{\prime}=A\mathbf{x}$.

Generalization

Let $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$ be functions of $t$. Let $F_{1}$, $F_{2}$, $\cdots$, $F_{n}$ be functions of $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$. Then, the system of first-order differential equations for $x_{i}(t),$ $1\le i \le n$ is as follows.

$$\begin{align*} x_{1}^{\prime}(t) =&\ F_{1}(t,x_{1},x_{2},\cdots,x_{n}) \\ x_{2}^{\prime}(t) =&\ F_{2}(t,x_{1},x_{2},\cdots,x_{n}) \\ \vdots & \\ x_{n}^{\prime}(t) =&\ F_{n}(t,x_{1},x_{2},\cdots,x_{n}) \end{align*} \tag{1}$$

When each $F_{i}$ is linear, it is called a linear system, and if not, it is called a nonlinear system. The more general form of a first-order linear differential equation system is as follows.

$$\begin{align*} x_{1}^{\prime}(t) =&\ p_{11}(t)x_{1}(t)+\cdots p_{1n}(t)x_{n}(t) + g_{1}(t) \\ x_{2}^{\prime}(t) =&\ p_{21}(t)x_{1}(t)+\cdots p_{2n}(t)x_{n}(t) + g_{2}(t) \\ \vdots & \\ x_{n}^{\prime}(t) =&\ p_{n1}(t)x_{1}(t)+\cdots p_{nn}(t)x_{n}(t) + g_{n}(t) \end{align*}$$

$$\mathbf{x}^{\prime}(t) = \mathbf{P}(t)\mathbf{x}(t) + \mathbf{g}(t)$$

Here, $\mathbf{x}$, $\mathbf{g}$ are vector-valued functions, and $\mathbf{P}$ is a matrix function. If each $g_{i}(t)$ is $0$, it is called a homogeneous system; otherwise, it is called a nonhomogeneous system.

Solutions

The solution to the ODE system $(1)$ over interval $I : \alpha \lt t \lt \beta$ consists of $n$ functions that are differentiable at each point on interval $I$.

$$x_{1} = \phi_{1}(t),\quad x_{2} = \phi_{2}(t),\quad \dots,\quad x_{n} = \phi_{n}(t)$$

Initial Conditions

For fixed $t_{0} \in I$ and the $x_{i}^{0}$s, the following $n$ conditions are called initial conditions.

$$x_{1}(t_{0}) = x_{1}^{0},\quad x_{2}(t_{0}) = x_{2}^{0},\quad \cdots,\quad x_{n}(t_{0}) = x_{n}^{0} \tag{2}$$

Combining the ODE system $(1)$ with initial conditions $(2)$ is called the initial value problem, commonly abbreviated as IVP. ‘Finding the solution to the initial value problem’ is referred to as ‘solving the initial value problem’. The solution to the initial value problem is guaranteed to exist and be unique by Picard’s theorem.