Calculus // Systems Over Time

DIFFERENTIAL
EQUATIONS

If you know how fast a car is accelerating, you can calculate exactly where it will be in ten minutes. A differential equation defines the rules of change for a system, allowing us to mathematically predict the future.

The Language of Nature

In algebra, you solve an equation to find a specific number, like $x = 5$ . In differential equations, you solve an equation to find an entire function.

We write these equations by relating a function $y$ to its own derivative $\frac{dy}{dx}$ . For example, Newton's Law of Cooling states that the rate an object cools down is proportional to the difference between its temperature and the room temperature:

\frac{dT}{dt} = -k(T - T_{room})

Initial Value Problems

A differential equation does not give you a single answer. It gives you a "Slope Field"—a map of how the system would behave if it started at any given point.

To find out what actually happens, you must provide an Initial Condition

(x_0, y_0)

. Once you drop that mathematical pin, the equation locks into a single, inescapable destiny.

Try selecting the Logistic equation in the lab. This models population growth. No matter where you click (even if you start with an overpopulation at the top), the trajectory will always eventually flatline at the carrying capacity $y = 4$ .

Slope Field Integrator

Euler's Method Approximation

Click to drop an Initial Value

Initial Value Problem: The teal lines represent the derivative dy/dx. Click anywhere to set an initial condition (x₀, y₀). The engine will instantly integrate the mathematical flow to reveal the singular destiny of that specific particle.

Nonlinear Dynamics

What happens when you have multiple equations interacting with each other simultaneously?

DIFFERENTIAL
EQUATIONS

The Language of Nature

Initial Value Problems

Slope Field Integrator

Nonlinear Dynamics

Systems of ODEs

Partial Differential Equations

Chaos Theory