Systems of ODEs

Until now we have considered ODEs with one unknown function, but it is often useful to consider systems of ordinary differential equations, where we have multiple unknown functions, satisfying multiple relations between themselves and their derivatives. A prototypical example of a system of ODEs comes from predator-prey models, where predator and prey populations satisfy a system of coupled differential equations that govern their time evolution. Here is an example.

${\displaystyle {\begin{cases}{\frac {dx}{dt}}=x-xy\\{\frac {dy}{dt}}=-y+xy\end{cases}}}$

In this case, the two unknown functions are ${\displaystyle x(t)}$, representing the prey population, and ${\displaystyle y(t)}$, representing the predator population. Another reason for us to care about systems of ODEs is that any ODE of arbitrarily high order, with arbitrarily many different unknown functions, can be written as a system of first order ODEs. This is a tremendously useful fact, because, at least theoretically, it reduces the study of ordinary differential equations to the study of first-order ordinary differential systems, which simplifies analysis considerably. Here is how this works. We present the case of one nth order linear homogenous ODE, but case of a nonlinear ODE and a higher order system are easy extensions. Suppose we have an ODE of the following form:

${\displaystyle {\frac {d^{n}y}{dt^{n}}}+a_{n-1}{\frac {d^{n-1}y}{dt^{n-1}}}+\cdots +a_{0}y=0}$

Define the following variables:

${\displaystyle {\begin{aligned}x_{1}&=y\\x_{2}&={\frac {dy}{dt}}\\&\vdots \\x_{n}&={\frac {d^{n-1}y}{dt^{n-1}}}\end{aligned}}}$

Then, we have ${\displaystyle {\frac {dx_{1}}{dt}}={\frac {dy}{dt}}=x_{2}}$, and similarly, ${\displaystyle {\frac {dx_{2}}{dt}}={\frac {d^{2}y}{dt^{2}}}=x_{3}}$. Proceeding in this fashion, we can compute the derivatives of all the new variables, and we obtain the following system:

${\displaystyle {\begin{cases}{\frac {dx_{1}}{dt}}=x_{2}\\{\frac {dx_{2}}{dt}}=x_{3}\\\qquad \vdots \\{\frac {dx_{k}}{dt}}=x_{k+1}\\\qquad \vdots \\{\frac {dx_{n}}{dt}}=-a_{n-1}x_{n}-a_{n-2}x_{n-1}-\cdots -a_{0}x_{1}\end{cases}}}$

This is exactly a first order system that's equivalent to the original higher order ODE. A few comments are in order: First of all, solving this system simultaneously solves for the unknown function ${\displaystyle y}$ and its derivatives. In practice, all this information may not be needed, so this justifies having separate, direct methods for solving higher order ODEs. Second of all, the assumption that our ODE is linear and homogenous allows us to rewrite our system in a nice form as follows:

Define

${\displaystyle \mathbf {x} ={\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}}$

and

${\displaystyle A={\begin{pmatrix}0&1&\cdots &\cdots &0\\0&0&1&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\-a_{0}&-a_{1}&\cdots &\cdots &-a_{n-1}\end{pmatrix}}.}$

Then, using standard conventions for matrix-vector multiplication, we can write our system as

${\displaystyle {\frac {d\mathbf {x} }{dt}}=A\mathbf {x} .}$

First of all, this system should look suspiciously similar to the scalar equation ${\displaystyle {\frac {dx}{dt}}=ax}$. This well-known equation is solved by the exponential function, which might suggest how one might approach solving the system. Second of all, this matrix-vector form allows us to apply linear algebra to solving differential equation, and it will turn out that the properties of the original ODE will depend heavily on the properties of the matrix ${\displaystyle A}$. These topics will be explored later.