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Imagine you are tracking a weather balloon, but your instrument can only measure its speed and slope at each coordinate — not its actual path. How do you reconstruct where it went?
A differential equation is an equation that connects an unknown function yy to its rate of change, the derivative dydx\frac{dy}{dx}.
The order of a differential equation is the highest derivative that appears in it. When only the first derivative dydx\frac{dy}{dx} appears, we call it a first-order differential equation.
In the 1670s, Gottfried Wilhelm Leibniz and Isaac Newton developed these equations to describe how physical systems continuously change over time.
When we reverse differentiation to solve dydx=f(x)\frac{dy}{dx} = f(x), what does the unknown constant of integration actually do to the graph?

The General Solution and Families of Curves

Because the derivative of any constant number is zero, integrating a first-order differential equation introduces an arbitrary constant +C+ C.
Consider the simple differential equation dydx=2x\frac{dy}{dx} = 2x. Integrating both sides with respect to xx gives the general solution y=x2+Cy = x^2 + C.
Geometrically, this solution is not just one curve — it is a family of curves, where each distinct value of CC produces a different curve that shares the exact same slope rule.
Does changing CC always just slide curves up and down, or can it change their shape and direction entirely?

Nonlinear Families: Scaling and Reflection