Modeling Situations with Differential Equations

Many real situations describe how fast something changes rather than its value outright. 'The population grows at a rate proportional to its size' or 'the coffee cools at a rate proportional to how far it is from room temperature' are both sentences about rate of change, not about a specific number. Since rate of change means derivative, each sentence converts directly into an equation with a derivative in it. 'Proportional to' means 'equals a constant times,' so 'the rate of change of y is proportional to y' becomes dy/dt = ky, where k is the constant of proportionality. If the quantity is decreasing, k is negative; if growing, k is positive.

The key shift is that the unknown in a differential equation is not a number but a whole function, y(t), that makes the equation true for every t. Writing down dy/dt = ky is just the translation step; it does not tell you what y actually is as a function of t. Finding that function is a separate job, done later by separating variables and integrating. That is why plugging in numbers and solving for y algebraically, the way you'd solve 3x + 5 = 20, does not work here: there is no single y to isolate, only a relationship between y and its own rate of change.

This translation step is the seed for everything else in the unit. Once you accept dy/dt = ky as the model, you can draw slope fields showing the pattern of slopes it predicts, sketch solution curves that follow those slopes, and eventually solve the equation to recover the exponential growth or decay formula y = Ce^kt. Newton's law of cooling, dy/dt = -k(y - R), is the same idea with the target value shifted to room temperature R instead of zero.