A differential equation is an equation that relates one or more functions and their derivatives. For example, the equation dy dx equals 2x relates the function y of x to its derivative. The solution to this differential equation is y equals x squared, which we can see plotted here.
There are two main types of differential equations. Ordinary differential equations, or ODEs, involve functions of one variable and their derivatives. For example, the second derivative of y plus 3 times the first derivative plus 2y equals zero. Partial differential equations, or PDEs, involve functions of multiple variables and their partial derivatives, like Laplace's equation shown here.
Differential equations are classified by their order and degree. The order is the highest derivative present in the equation. For example, dy dx equals 2x is first order, while the second derivative of y plus y equals zero is second order. The degree is the power of the highest order derivative. Here we see solutions to first and second order equations plotted together.
Differential equations have numerous applications across many fields. In physics, they describe motion and oscillations, like the simple harmonic oscillator equation. In biology, they model population growth and disease spread using equations like the logistic growth model shown here. In economics, they help analyze market dynamics and financial systems.
To summarize what we have learned about differential equations: They are mathematical equations that relate functions to their derivatives. There are two main types, ordinary and partial differential equations. They are classified by their order and degree. Differential equations have wide applications across science and engineering, serving as essential tools for modeling dynamic systems in our world.