Differential Equations · Chapter 02

Separable and autonomous equations

The first family of equations that can be solved outright, and the first that can be understood without being solved. Separation turns an equation into two integrals; the phase line reads the long run off the sign of the right-hand side alone.

y^{'} = g (t) h (y)

y^{'} = f (y)

\int \frac{d y}{h ( y )}

y^{'} = k y

y^{'} = r y (1 - \frac{y}{K})

f (y_{*}) = 0

y = C e^{k t}

y_{*}

Lessons

Build the foundation

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05Phase linesThe direction field of an autonomous equation repeats sideways, so all the information in it lives on a single vertical line. Mark the equilibria on that line and put an arrow on each interval between them, pointing up where the rate is positive and down where it is negative. That one picture predicts the fate of every solution of the equation, and drawing it takes no integration at all.Read lesson →

06Stability of equilibriaAn equilibrium is stable if a small disturbance dies away and unstable if it grows. The phase line answers the question by inspection: arrows pointing in on both sides mean stable, arrows pointing out mean unstable, and one of each means the equilibrium attracts from one side and repels from the other. There is also a test in one derivative, which is the version that survives into the chapter on systems.Read lesson →

07Exponential growth and decayThe simplest interesting autonomous equation says the rate is proportional to the amount. Separating it takes two lines and gives the exponential, and the sign of the constant decides everything: growth without bound one way, decay to nothing the other. The model is exact for radioactivity, good for money, and wrong for populations over any long stretch — which is what the next lesson repairs.Read lesson →

08The logistic equationA population cannot grow in proportion to itself for ever, because the habitat runs out. The cheapest honest repair is to multiply the growth rate by a factor that falls to zero as the population reaches what the habitat carries. The result is the most-studied equation in this chapter, and everything about it can be read from its phase line before a single integral is attempted.Read lesson →

09Chapter practiceBring the chapter together: separate the variables and integrate, find the equilibria of an autonomous equation and read its phase line, test an equilibrium for stability with one derivative, and use the exponential and logistic models. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Separable and autonomous equations

Build the foundation