mathematics//differential equations

An equation whose unknown is a function and whose statement is about its rate of change. The modeling idea behind almost all of them fits in one line: the derivative of a quantity with respect to time equals **what makes it increase minus what makes it decrease**.

An equation whose unknown is a function and whose statement is about its rate of change. The modeling idea behind almost all of them fits in one line: the derivative of a quantity with respect to time equals what makes it increase minus what makes it decrease.

Start with d(quantity)/dt = gains minus losses.

Then add the mechanisms one term at a time: an interaction term when two things must meet for something to happen, feedback when the quantity acts on its own rate, an external forcing when the world pushes on a schedule of its own, spatial diffusion when neighbors in space exchange the quantity.

Two families by the number of independent variables. An ordinary differential equation has one, almost always time, and its unknowns are functions of that one variable. A partial differential equation has several, typically space and time, and its unknown is a field.

What every one of them needs before it says anything concrete: an initial condition, because the equation alone has a whole family of solutions, and, when the model has one, a constraint the solutions must respect.

The equation is the rule; what the rule produces over time (trajectories, equilibria, modes, stability) is the subject of dynamical systems. The simplest complete example, a proportional rate, is exponential decay; the simplest second-order one is the oscillator.