Systems of First Order Ordinary Differential Equations

# Systems of First Order Ordinary Differential Equations

Recall from the First Order Ordinary Differential Equations page that if $D \subseteq \mathbb{R}^2$ is a domain (a nonempty, open, connected subset of $\mathbb{R}^2$) and $f \in C(D, \mathbb{R})$ then a first order ordinary differential equation has the form:

(1)
\begin{align} \quad x' = f(t, x) \end{align}

Furthermore we said that a solution to a first order ordinary differential equation on an interval $J = (a, b)$ is a function $\phi \in C^1(J, \mathbb{R})$ (continuous, differentiable function) such that for all $t \in J$ we have that $(t, \phi(t)) \in D$ and:

(2)
\begin{align} \quad \phi'(t) = f(t, \phi(t)) \end{align}

We defined a solution to the initial value problem $x' = f(t, x)$ on $J$ with initial condition $x(\tau) = \xi$ where $\tau \in J$ is a solution to $\phi$ to $x' = f(t, x)$ as above with the added condition that $\phi(\tau) = \xi$.

We are now ready to define a system of first order ordinary differential equations.

 Definition: Let $D \subseteq \mathbb{R}^{n+1}$ be a domain and let $f_1, f_2, ..., f_n \in C(D, \mathbb{R})$. A System of $n$ First Order Ordinary Differential Equations is of the form $\displaystyle{\left\{\begin{matrix} x_1' = f_1(t, x_1, x_2, ..., x_n)\\ x_2' = f_2(t, x_1, x_2, ..., x_n)\\ \vdots \\ x_n' = f_n(t, x_1, x_2, ..., x_n) \end{matrix}\right.}$, sometimes abbreviated as $\{ x_i' = f_i(t, x_1, x_2, ..., x_n) \}$ where $i \in \{ 1, 2, ..., n \}$.

Remember once again that we say that $D$ is a domain of $\mathbb{R}^{n+1}$ if $D$ is a nonempty, open, connected, subset of $\mathbb{R}^{n+1}$.

The following is an example of a system of 4 first order differential equations:

(3)
\begin{align} \quad x_1' &= tx_1 + t^2x_2 + t^3x_3 + t^4x_4\\ x_2' &= (t + 1)x_1x_4\\ x_3' &= x_1^3 + 1\\ x_4' &= x_3 + \cos (t)x_4 \end{align}

Here we have that $f_1, f_2, f_3, f_4 : \mathbb{R}^{n+1} \to \mathbb{R}$ are given explicitly by:

(4)
\begin{align} \quad f_1(t, x_1, x_2, x_3, x_4) &= tx_1 + t^2x_2 + t^3x_3 + t^4x_4 \\ \quad f_2(t, x_1, x_2, x_3, x_4) &= (t + 1)x_1x_4 \\ \quad f_3(t, x_1, x_2, x_3, x_4) &= x_1^3 + 1 \\ \quad f_4(t, x_1, x_2, x_3, x_4) &= x_3 + \cos (t)x_4 \end{align}

Note that while the functions $f_1, f_2, f_3, f_4$ are functions of $t, x_1, x_2, x_3, x_4$ that they need not depend on all of those variables.

 Definition: A Solution to a system of $n$ first order ordinary differential equations on an open interval $J = (a, b)$ is a collection of continuously differentiable functions $\phi_1, \phi_2, ..., \phi_n \in C(J, \mathbb{R})$ such that for all $t \in J$ we have that $(t, \phi_1(t), \phi_2(t), ..., \phi_n(t)) \in D$ and $\displaystyle{\left\{\begin{matrix} \phi_1'(t) = f_1(t, \phi_1(t), \phi_2(t), ..., \phi_n(t))\\ \phi_2'(t) = f_2(t, \phi_1(t), \phi_2(t), ..., \phi_n(t))\\ \vdots \\ \phi_n'(t) = f_n(t, \phi_1(t), \phi_2(t), ..., \phi_n(t)) \end{matrix}\right.}$ sometimes abbreviated $\{ \phi_i'(t) = f(t, \phi_1(t), \phi_2(t), ..., \phi_n(t) \} where [[$ i \in \{ 1, 2, ..., n \}$.  Definition: An Initial Value Problem is a system of$n$first order ordinary differential equations$\{ x_i' = f(t, x_1, x_2, ..., x_n) \}$,$i \in \{ 1, 2, ..., n \}$with initial conditions$\{ x_i(\tau) = \xi_i \}$,$i \in \{ 1, 2, ..., n \}$where$(\tau, \xi_1, \xi_2, ..., \xi_n) \in D$. A Solution to the initial value problem$\{ x_i' = f(t, x_1, x_2, ..., x_n) \}$,$\{ x_i(\tau) = \xi_i \}$,$i \in \{ 1, 2, ..., n \}$on the open interval$J = (a, b)$with$\tau \in J$is a solution$\{ \phi_1, \phi_2, ..., \phi_n \}$to$\{ x_i' = f(t, x_1, x_2, ..., x_n) \}$with$\{ \phi_i(\tau) = \xi_i \}$for all$i \in \{ 1, 2, ..., n \}\$.