Calculus and modelling

A differential equation describes directions before it describes a curve.

This prototype introduces slope fields for first-order differential equations. Learners can choose an equation, set an initial condition and watch a numerical solution follow the local gradients.

The aim is to connect calculus, modelling and numerical methods in a way that feels visual rather than purely symbolic.

Differential equations lab

Read a differential equation as a field of directions

A first-order differential equation tells you the gradient at each point. The slope field shows those local directions, and the curve follows them from the chosen initial condition.

Equation

dy/dx = 0.7(1 - y)

Initial point

(-3, 2.2)

Initial gradient

-0.84

Step size

0.08

Controls

Equation

Initial x: -3.0

Initial y: 2.2

Numerical step size: 0.08

dy/dx = 0.7(1 - y)

Solutions are pulled towards y = 1, like a simplified cooling or adjustment model.

The equilibrium solution is y = 1.

What this shows

A differential equation can describe a whole family of possible curves before any initial condition is chosen.

The initial condition selects one particular solution from that family.

Numerical methods follow the local gradient repeatedly, which is why step size matters.

Guided tasks

Hide the solution curve. Predict the direction from the initial point, then reveal it.

Find an initial condition that approaches an equilibrium solution.

Increase the step size. Does the numerical curve still look trustworthy?

Joy in the process

The point is not just to finish. It is to notice, test and return.

These tools are invitations to explore. A good mistake, a surprising pattern or a question you cannot yet answer is part of the work, not a failure of it.

The challenge is deliberate: the site should support thinking, not remove the need for it.

Before changing a setting, pause and predict what you think will happen.

Change one thing at a time. What stayed the same, and what changed?

Try to create a surprising case, a broken case, or a beautiful pattern.

Ask what this connects to outside the page: maps, movement, nature, systems or decisions.

Reset, then try again with a new question in mind.

Future extensions

This can grow into a full differential equations and modelling lab.

Add Euler method vs Runge-Kutta comparison.

Add exact solutions for selected equations where possible.

Add phase line views for autonomous differential equations.

Add modelling contexts such as cooling, population growth and epidemics.