Differential Equations

In this chapter, we discuss what probably is the most important application of calculus: differential equations. These equations are those that describe many laws of nature. In classical physics, we have to mention Newton’s second law \(F=m\ddot{x}\) that describes motions, Fourier’s heat equation \(\dot{\theta } = \kappa ^2 \nicefrac {\partial ^2\theta }{\partial x^2}\) that describes how heat is transferred in a medium, Maxwell’s equations describing electromagnetism and the Navier-Stokes equation that calculates how fluids move. In quantum mechanics, we have the Schrödinger equation. In biology, we can cite the Lotka-Volterra equations, also known as the predator-prey equations—a pair of first-order nonlinear differential equations—used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. In finance, there is the Black-Scholes equation.