Glossary
Differential equations
Equations that relate a function to its derivatives, describing how a quantity changes with respect to one or more independent variables.
Example:
The rate at which a cup of coffee cools can be modeled by a differential equation that relates its temperature to the rate of change of its temperature.
Euler's method
A numerical technique used to approximate the solution to a differential equation with a given initial condition by taking small, iterative steps along tangent lines.
Example:
To estimate the position of a projectile after 3 seconds given its initial velocity and a differential equation for its motion, you could apply Euler's method with a small time step.
Logistic differential equation
A specific type of differential equation that models population growth where the growth rate slows as the population approaches a maximum carrying capacity.
Example:
The spread of a new viral trend on social media might initially grow exponentially but then follow a logistic differential equation as it reaches saturation among users.
Mathematical modeling
The process of using mathematical concepts and language to describe, analyze, and predict the behavior of real-world phenomena.
Example:
Engineers use mathematical modeling to simulate airflow over an airplane wing, predicting its lift and drag without building a physical prototype.
Slope field
A graphical representation of a first-order differential equation, where small line segments are drawn at various points (x,y) to indicate the slope of the solution curve at that point.
Example:
By sketching a slope field for dy/dx = y - x, you can visually observe how different initial conditions lead to distinct solution curves, some increasing and some decreasing.