Types of first-order dynamical systems

Ordinary differential equations: Introduction

Types of first-order dynamical systems

Homogeneous and non-homogeneous linear ODE Let us first consider only first-order dynamical systems that can be written in the following explicit form: \[\frac{\dd y}{\dd t} = \varphi(t,y)\] This differential equation is linear when \[\varphi(t,y)=a(t)\cdot y(t)+b(t)\] where the functions \(a(t)\) and \(b(t)\) are the coefficients of the linear differential equation and \(b\) is the non-homogeneous term. Note that the coefficients of the linear differential equation do not necessarily have to be constant. If \(b(t)=0\) for all \(t\), then the linear differential equation is called homogeneous, Otherwise one speaks of a non-homogeneous or inhomogeneous differential equation.

For a linear ODE of order greater than 1 we can distinguish homogeneous and inhomogeneous ODEs.

A linear homogeneous ODE of order \(n\) can be written as \[a_n(t)\cdot \frac{\dd^n y}{\dd t^n}+a_{n-1}(t)\cdot \frac{\dd^{n-1} y}{\dd t^{n-1}}+\cdots +a_2(t)\cdot \frac{\dd^2 y}{\dd t^2}+a_1(t)\cdot \frac{\dd y}{\dd t}+a_0(t)\cdot y(t)=0\]
A linear inhomogeneous ODE of order \(n\) can be written as \[a_n(t)\cdot \frac{\dd^n y}{\dd t^n}+a_{n-1}(t)\cdot \frac{\dd^{n-1} y}{\dd t^{n-1}}+\cdots +a_2(t)\cdot \frac{\dd^2 y}{\dd t^2}+a_1(t)\cdot \frac{\dd y}{\dd t}+a_0(t)\cdot y(t)=b(t)\] for some nonzero function \(b(t)\).

In short, put every term in the ODE in which \(y\) and its derivative occur on the left-hand side of the equation and every other term to the right of the equal sign. Then check whether the right-hand side of the equation is equal to 0 or not.

Separable first-order ODE One speaks of a separable first-order ODE when it can be written in the form \[\frac{\dd y}{\dd t} = \frac{f(t)}{g(y)}\] for certain functions \(f\) and \(g\).
Moreover, if \(f(t)=1\) for all \(t\), then it is an autonomous first-order ODE.

The table below shows examples of the various types of first-order dynamical systems.

Examples \[\begin{array}{|l|l|} \hline \text{non-linear} & t\,y'=y\, \ln(t\,y)-y\phantom{x}\\ \hline \text{noninear, autonomous} & y'=\dfrac{-y}{\sqrt{1-y^2}} \\ \hline \text{non-linear, separable} & y'=\dfrac{t^2}{y} \\ \hline \text{linear, non-homogeneous} & y'=t\,y+\dfrac{1}{t^2} \\ \hline \text{linear, homogeneous} & y'=t\,y \\ \hline \end{array}\]

For simple ODEs, the probability is greater that one can find an analytical solution, i.e., a mathematical formula. Therefore, special attention is paid to this type of ODEs. Note that for more complicated ODEs, it is very common that you cannot find an exact, analytical solution! Most differential equations do not have an analytical solution! In that case, one uses numerical methods (e.g., Euler's method or Runge-Kutta methods) to determine numerical solutions and one applies bifurcation analysis to explore how behaviour of solutions depends on values of parameters in the differential equation.