Suppose a mass m, which is attached to a spring which is fixed to a wall, is resting on a table. Let $x\=0$ be the point where the mass is in rest position. If the mass is moved so that the spring is stretched or compressed, a force is applied to the mass.  Hooke's law states that the force exerted by the spring on the mass is $-k x$ where $k$ is the spring constant, that is, the force is proportional to the displacement $x$ of the mass from the rest position.  Since  $f\=m a$, that is, force equals mass times acceleration, we have the following second order differential equation which governs the motion of the mass.

$-kx\=m \frac{{\ⅆ}^{2}x}{{\ⅆ}^{}{t}^2}$

When the mass slides over the table there will be a frictional force applied to the mass in the opposite direction of motion.  Assuming it is proportional to the velocity of the mass, we obtain the following equation

   $\frac{{\ⅆ}^{2}x}{{\ⅆ}^{}{t}^2}\+\mathrm{\mu } \frac{\ⅆx}{\ⅆt}\+{\mathrm{\ω}}^{2}x\=0$ 

which governs the motion of the mass.  Here  $\mathrm{\omega }\=\sqrt{\frac{k}{m}}$ , $k$  is the constant of spring, $m$ is the mass.  The term $\mathrm{\mu } \frac{\ⅆx}{\ⅆt}$ models the friction with the table where $\mathrm{\mu }$ is the damping coefficient. One can convert this linear second order differential equation into a system of two first order differential equations by letting $v\=\frac{\ⅆx}{\ⅆt}$, that is, v is the velocity, obtaining:

  $\frac{\ⅆv}{\ⅆt}\+\mathrm{\mu }v\+{\mathrm{\omega }}^{2}x\=0$, $v\=\frac{\ⅆx}{\ⅆt}$

This is a two-compartment model for the metastasis of cancer, that is, the spread of cancer cells through the boundary of an organ tissue.  Let $x_1$ be the number of arrested cells (sitting on the boundary of the tissue in compartment A) and $x_2$ be the number of cells that have invaded the target tissue (compartment B). Cells pass from compartment A to B with a rate of ${\mathrm{\beta }}_2{x}_1$. In addition, cells die from compartment A with the rate ${\mathrm{\beta }}_1{x}_1$ and die from compartment B with the rate ${\mathrm{\beta }}_3{x}_2$.  We obtain the following system of two first order differential equations representing these relations.

 $\frac{\ⅆ\mathrm{x\_\_1}}{\ⅆt}\=-\left({\mathrm{\beta }}_1\+{\mathrm{\beta }}_2\right)x_1$, $\frac{\ⅆ\mathrm{x\_\_2}}{\ⅆt}\={\mathrm{\beta }}_2{x}_1-{\mathrm{\beta }}_3{x}_2$.

The Lotka-Volterra system

 $\frac{\ⅆ}{\ⅆt}x\left(t\right)\=\mathrm{\alpha }x\left(t\right)-bx\left(t\right)y\left(t\right)$ , $\frac{\ⅆ}{\ⅆt}y\left(t\right)\=-\mathrm{\beta }y\left(t\right)\+cx\left(t\right)y\left(t\right)$,

is a predator-prey model.  $x\left(t\right)$ is the population of the prey at time $t$, $y\left(t\right)$ is the population of the predator at time $t$, $0<\mathrm{\alpha }$ is the birth rate of the prey, $0<\mathrm{\beta }$ is the death rate of the predator, b is the predation rate coefficient, and c is the reproduction rate of predators per one prey eaten.

The brusselator is a mathematical model for chemical oscillation. The dynamics of this system are given by:

$\frac{\&DifferentialD;u}{\&DifferentialD;t}\&equals;1-\left(b\&plus;1\right)u\&plus;a{u}^2\mathrm{\nu }$ , $\frac{d\mathrm{\nu }}{\mathrm{dt}}\&equals;bu-a{u}^2\mathrm{\nu }$,

where $0\le u$ and $0\le \mathrm{\nu }$, are concentrations of two chemicals, and $a$ and $b$ are positive constants.
For certain values of a and b, for example, a = 0.5, b = 2, the brusselator exhibits natural oscillation, that is, a stable limit cycle.

Kermack-McKendrick model is an SIR (Susceptible, Infected, Recovered) model for the number of people infected with a contagious illness in a closed population over time.
Here we are using a simplified version of the model with no recovery, R, element. So we have

$\frac{\&DifferentialD;S}{\&DifferentialD;t}\&equals;-\mathrm{\beta }SI$,  $\frac{\&DifferentialD;I}{\&DifferentialD;t}\&equals;\mathrm{\beta }SI-\mathrm{\gamma }I$,

where 0 < S < 1 is the proportion of the population which is susceptible to the illness, 0 < I < 1 is the proportion of infectious individuals in the population,  $\mathrm{\beta }$ is the mean transmissions rate, and $\mathrm{\gamma }$ is the mean rate at which infected individuals either recover or die from the illness.

The most interesting observation to make about this model is that the S does not drop to zero, that is, not everyone in the population gets the illness, even if $\mathrm{\beta }$ is increased (individuals are more infectious) and $\mathrm{\gamma }$ is decreased (individuals stay infectious for longer).  In the plot above we see that when I=0, that is, when all infectious individuals have either died or recovered, the value of S=0.2, that is, about 20% of the population is susceptible, i.e., has not caught the illness.  In the plot below it is about S=0.02 or 2%.  This correctly predicts what has been observed in real life.

Suppose in a closed environment there are two species competing for a limited supply of food. Let $x$ and $y$ be the population of the two species at time $t$. We have

 $\frac{\&DifferentialD;x}{\&DifferentialD;t}\&equals;x\left(a-{\mathrm{\delta }}_{1}x-{\mathrm{\alpha }}_{1}y\right)$ ,  $\frac{\&DifferentialD;y}{\&DifferentialD;t}\&equals;y\left(b-{\mathrm{\delta }}_{2}y-{\mathrm{\alpha }}_{2}x\right)$,

where $a$ and $b$ are the growth rates of the two populations, and $\frac{a}{{\mathrm{\delta }}_1}$ and $\frac{b}{{\mathrm{\delta }}_2}$ are their saturation levels. The $\mathrm{\alpha }$ values are a measure of the degree to which each species interferes with the other.

The van der Pol equation describes the self-sustaining oscillations in which the energy is fed into small oscillations and removed from large oscillations. This equation arises in the study of circuits containing vacuum tubes. The Van der Pol equation is given by:

$\frac{{\&DifferentialD;}^{2}x}{{\&DifferentialD;}^{}{t}^2}-\mathrm{\&mu;} \left(1-x^2\right) \frac{\&DifferentialD;x}{\&DifferentialD;t}\&plus;x\&equals;0$,

where $\mathrm{\mu }$, a positive constant, describes the current $x$ in a triode oscillator. When $\mathrm{\mu }\&equals;0$ the equation reduces to the simple harmonic oscillator. One can write this second order differential equation as a system of differential equations:

$\frac{\&DifferentialD;}{\&DifferentialD;t}x\left(t\right)\&equals;y\left(t\right)$, $\frac{\&DifferentialD;}{\&DifferentialD;t}y\left(t\right)\&equals;-x\left(t\right)\&plus;\mathrm{\mu }\left(1-{x\left(t\right)}^2\right)y\left(t\right)$.

The following system of differential equations describes the relationship between temperatures of rooms in a house with a furnace.
 A(t) is the temperature for room A, B(t) is the temperature for room B, F is the heating rate of the furnace located in room B, and Am is the outside temperature.

$\frac{\&DifferentialD;}{\&DifferentialD;t}A\left(t\right)\&equals;k_1\left(\mathrm{Am}-A\left(t\right)\right)\&plus;k_2\left(B\left(t\right)-A\left(t\right)\right)$, $\frac{\&DifferentialD;}{\&DifferentialD;t}B\left(t\right)\&equals;k_2\left(A\left(t\right)-B\left(t\right)\right)\&plus;k_3\left(\mathrm{Am}-B\left(t\right)\right)\&plus;F$

The parameters $\mathrm{k1}\&comma;\mathrm{k2}$, and k3 describe the rates of exchange for heat between the rooms A,and B,and outside.

Consider two competing nations. Both nations are self defensive and fight back to protect their nation. Both nations maintain army and stock weapons. When one nation expands their army the other nation finds it offensive. Let x(t) and y(t) represent the yearly rate of armament expenditures of the two nations in some standard unit. One can model the Mutual Fear factor for each nation. Assume that extensive armament expenditures create a drag on the nation's economy. And also assume that the each nation's mutual fear's rate is directly proportional to the expenditure of the other nation. In addition, assume that there are constants r and s indicating the grievance of one country toward the other country. Note that negative values for s and r represent feelings of good will.
The Model will be:

$\frac{\&DifferentialD;x}{\&DifferentialD;t}\&equals;ay-mx\&plus;r\&comma;\frac{\&DifferentialD;y}{\&DifferentialD;t}\&equals;bx-ny\&plus;s$

where a, b, m, and n are positive constant.