Ordinary Differential Equations/ODEs as Models

Ordinary Differential Equations naturally arise in a wide variety of applied fields as models of real phenomena. Differential equations are ubiquitous in physics, engineering, chemistry, economics, and many other areas. Sometimes it is difficult to find an algebraic equation that directly contains some quantity of interest, but modeling principles may lead to a differential equation satisfied by the desired quantity.

One of the first ever differential equations, and one of the most important in applications, is Newton's Second Law, $F=m{\frac {dv}{dt}}$, where $F$ is the magnitude of a force applied to a particle, $m$ is the mass of the particle, and $v$ is the particle's velocity in some inertial reference frame. Many common ODEs of interest end up being Newton's Second Law applied to some physical system. One example of this is the damped harmonic oscillator, whose dynamics are governed by $m{\ddot {x}}+b{\dot {x}}+kx=0$, where $m$ is the particle position, $b$ is the damping coefficient, and $k$ is the spring constant.

However, one of the most remarkable facts about modeling with ODEs is that most of the ODEs that admit explicit solutions are widely applicable in many areas. For example, the damped harmonic oscillator is analogous to a model that is often used in electrical circuits, and it is usually applicable to any application that has some appropriate energy storage and dissipation elements.

Another common example of an ODE with many applications is the ODE for exponential growth and decay, ${\frac {dx}{dt}}=cx$, where $c>0$ corresponds to growth, and $c<0$ corresponds to decay. Exponential growth is often used to model populations in biology, like bacteria on a petri dish. It can also be used to describe a limiting case of saving with compound interest, where interest compounds continuously rather than at fixed time steps. Exponential decay is a common model for radioactive decay, where the amount of a radioactive substance in a sample decays exponentially over time. The same equation can be used to model diffusion of a solute in a fluid.

In summary, differential equations commonly arise as models of real world situations, and this is one of the prime motivations for studying them.