The rule for how things change
A cooling cup of coffee doesn't drop to room temperature at a fixed rate — the hotter it is, the faster it cools. A differential equation captures exactly that: an equation not about a value, but about how fast the value is changing.
An equation about change
A differential equation relates a function to its own rate of change. Instead of solving for a number, you solve for a whole function. For example, y' = y says 'the rate of growth equals the current amount' — its solution is the exponential y = Ceˣ.
Slope fields show the flow
At each point you can draw a tiny line with the slope y' demands. Solutions are curves that follow those little slopes everywhere — like leaves drifting on a current.
Check y = e^{2x} in y' = 2y: y' = 2e^{2x} and 2y = 2e^{2x}. They match, so it is a solution.