Equations That Track Change
A differential equation is a rule for how a quantity changes, and solving it means recovering the quantity itself from that rule.
What a solution is
A solution is a function whose derivative satisfies the equation at every point in time. For example, y equals e to the negative 0.5 t solves the equation y prime equals negative 0.5 y, because differentiating gives back negative 0.5 times the function itself. The order of the equation is the highest derivative that appears in it.
\frac{dy}{dt} = -0.5yFrom story to equation
Modeling means translating a sentence about rates into symbols. Coffee cooling becomes dT/dt equals negative k times the difference between T and room temperature, because cooling is proportional to how far the coffee is from room temperature. The same proportional-reasoning pattern gives interest growth, drug decay, and population drift models.
\frac{dT}{dt} = -k(T - A)Initial conditions
Solutions come in families, usually with one free constant for a first-order equation. An initial condition such as y at time 0 equals 100 selects exactly one member of the family. For the decay example, the general solution is C e to the negative 0.5 t, and the initial condition pins C to 100, giving a complete prediction.
y(t) = 100\,e^{-0.5t}Flashcard
General form of a first-order equation
Check your understanding
A cup of coffee sits in a 70-degree room, and its temperature drops at a rate proportional to the gap between the coffee and the room. Which differential equation models the coffee temperature T?