First-order equations: equilibria, existence and special methods

Lesson 1 solved first-order equations by separation of variables and integrating factors. Many first-order models cannot be solved that way, and even when they can, the formula may hide what an engineer most wants to know: does the system settle down, and to what? Will a small disturbance grow or die away? Is there exactly one solution, so that the model actually predicts something? This lesson takes a more qualitative view. It studies autonomous equations through their equilibria and phase lines, shows how equilibria can appear, disappear or change stability as a parameter changes, states the existence and uniqueness theorem and shows what happens when its conditions fail, and adds two further solution methods, exact equations and the Bernoulli substitution, which handle important nonlinear cases.

Autonomous equations and equilibria

An autonomous equation has a right-hand side that depends only on the unknown, not explicitly on time:

Newton's law of cooling, logistic growth, a tank draining through an orifice and many chemical rate laws are autonomous: the rules do not change with the clock. An equilibrium (or critical point) is a value \(y^*\) with \(f(y^*) = 0\); if the system starts there, it stays there forever. Between equilibria, \(f(y)\) has a constant sign, so solutions move steadily up (where \(f > 0\)) or down (where \(f < 0\)) and can never cross an equilibrium, because uniqueness (below) forbids two solutions from meeting.

All of this can be summarized on a phase line: a vertical line marked with the equilibria and arrows showing the direction of motion between them. An equilibrium is stable (an attractor) if nearby solutions move toward it from both sides, unstable (a repeller) if they move away on both sides, and semi-stable if they approach from one side and leave on the other. For a smooth \(f\), the slope at the equilibrium usually decides: \(f'(y^*) < 0\) means stable, \(f'(y^*) > 0\) unstable.

Bifurcations

Many models contain a parameter, such as a harvest rate, a feed rate or a supply voltage, and the number or stability of equilibria can change abruptly as it varies. Such a change is a bifurcation. In the harvested logistic model, raising the harvest \(h\) moves the two equilibria toward each other; at \(h = 12.5\) thousand per year they merge at \(P = 50\), and for any larger harvest there is no equilibrium at all: the population declines to zero from any starting value. This saddle-node bifurcation sets the maximum sustainable yield.

Bifurcations appear throughout engineering: a column that stays straight under increasing load until it suddenly buckles (Lesson 6), a chemical reactor that jumps between low- and high-temperature operating states, a power system whose voltage collapses when demand exceeds a critical value, and a fluid flow that changes from steady to oscillating as speed increases.

Existence and uniqueness

A model is useful only if it determines the future: from a given initial state, there should be exactly one solution. The existence and uniqueness theorem guarantees this under mild conditions. For the initial value problem \(y' = f(t, y)\), \(y(t_0) = y_0\): if \(f\) and its partial derivative \(\partial f/\partial y\) are continuous in a region around \((t_0, y_0)\), then there is exactly one solution, at least for some interval of time around \(t_0\).

The conditions are usually satisfied in engineering models, but two kinds of failure are instructive. First, the solution may exist only for a limited time: \(y' = y^2\) with \(y(0) = 1\) has the solution \(y = 1/(1 - t)\), which blows up at \(t = 1\), although the equation looks harmless. Models of runaway processes, such as thermal runaway in batteries, can show this kind of finite-time blow-up. Second, uniqueness can fail where \(\partial f/\partial y\) is infinite.

Exact equations

Some first-order equations come from differentiating a known function along a solution curve. Writing the equation as \(M(x, y)\,dx + N(x, y)\,dy = 0\), it is exact if there is a function \(F(x, y)\) with \(\partial F/\partial x = M\) and \(\partial F/\partial y = N\). Then the equation says \(dF = 0\), and the solutions are the level curves \(F(x, y) = C\). Since mixed partial derivatives are equal (MATH-209), a necessary and (in simple regions) sufficient test for exactness is:

Exact equations arise naturally from conservation laws: if a quantity such as energy is conserved, its level curves are the trajectories of the system.

When an equation is not exact, multiplying by a suitable integrating factor can sometimes make it so. The integrating factor of Lesson 1 for linear equations is one example of this idea.

The Bernoulli equation

An equation of the form \(y' + P(t)y = Q(t)y^n\), with \(n \ne 0, 1\), is nonlinear, but the substitution \(u = y^{1-n}\) turns it into a linear equation for \(u\), which the integrating factor method solves. Bernoulli equations describe, among other things, logistic growth, some drag laws and certain chemical kinetics.

Scaling to dimensionless form

Before analysing a model, it pays to rescale it so that the variables have no units. In the harvested logistic model, let \(p = P/K\) (population as a fraction of the carrying capacity) and \(\tau = rt\) (time measured in growth times). The equation \(P' = rP(1 - P/K) - h\) becomes \(dp/d\tau = p(1 - p) - H\), with the single dimensionless group \(H = h/(rK)\). Three parameters have collapsed into one, so a single phase line or bifurcation diagram covers every fishery: equilibria exist only for \(H \le 1/4\), which for r = 0.5 per year and K = 100 thousand gives the maximum sustainable harvest of 12.5 thousand per year found above. Dimensionless groups such as this one, and the Reynolds and Biot numbers in fluid mechanics and heat transfer, show which combinations of parameters actually matter and allow results from a laboratory model to be applied to a full-scale system.

Choosing an approach

For a first-order model, an engineer typically asks: Is it separable or linear (Lesson 1)? Is it exact, or a Bernoulli equation? If none of these, is it autonomous, so that a phase line shows its long-term behaviour? If a formula is needed anyway, numerical methods (MATH-300) will provide one. In every case, finding the equilibria and their stability first is cheap and tells you what the solution must do, which is the best check on any formula or simulation.