How are differential equations used to study motion and forces?

In mathematics and physics, differential equations are essential for studying motion and forces because they relate a system's state to its rates of change.1

According to Newton’s laws of motion, the force $F$ applied to a body of mass $m$ is proportional to its acceleration, expressed as $F=m{x}^{″}$, where $x^{″}$ is the second derivative of position with respect to time. Because acceleration is the rate of change of velocity, and velocity is the rate of change of position, determining an object's position $x(t)$ requires solving this differential equation.23

When forces are constant, such as in terrestrial gravity, these equations can be solved through integration. However, when forces depend on position or velocity, the equations become more complex, often requiring advanced analytical techniques or numerical analysis to predict the system's behavior. By establishing these mathematical models, scientists can describe how physical systems evolve over time, effectively bridging the gap between observed forces and the resulting motion of objects.2314