differential equation
Imagine you do not know how big something is at every moment, but you do know the RULE for how fast it changes. A savings account: you may not have a formula for the balance, but you know it grows by 3 percent of itself each year. A cooling cup of coffee: you do not know its temperature ten minutes from now, but you know it loses heat faster when it is hotter than the room. A differential equation is exactly this kind of statement — a rule about rates of change — written down as an equation.
Precisely, a differential equation is an equation that relates an unknown function to its own derivatives. The unknown is not a single number you solve for, as in 3x + 2 = 11; it is a whole function, say y(x), and the equation ties together y and its derivatives y', y'', and so on. For example y' = k y says 'the rate of change of y equals k times y itself.' Solving the equation means finding the function (or functions) that make the statement true everywhere, not finding a number.
This single idea is one of the most powerful tools in all of science, because nature almost always tells us its laws in the language of rates. Newton's second law, the spread of an epidemic, the decay of a radioactive sample, the swing of a pendulum, the flow of current in a circuit — every one of these begins as a differential equation. The whole subject is the art of turning a rule about change back into a description of the thing itself.
y' = k y is a differential equation: it says the slope of the unknown function y at each point equals k times the height of y there. If k > 0 the function grows ever faster; if k < 0 it decays toward zero.
The simplest growth/decay law — a rule about slope, not about value.
The solution is a function, not a number. Asking 'what is the answer' is asking 'which function (or which family of functions) satisfies the rule everywhere.'