The equation with only one answer

The two definitions so far in this ladder both start from a curve and derive a property. One takes the family $b^x$ and asks which base makes the slope equal the height; the other takes the area under a hyperbola and inverts it. Both arrive at $e$, and both begin by writing down a function.

The third route begins by writing down a condition and having the function forced out of it — which is the order in which the exponential actually turns up. Nobody observes an exponential curve. What is observed is that something changes at a rate proportional to how much of it there is.

What the field says #

Write $y'= ky$. At each point of the plane this says what the slope must be there, and nothing else. It does not say where a solution starts, how big it gets, or whether one exists.

Drawing the condition is a matter of putting a short segment at each point at the demanded slope, and the picture that results — a slope field — is the differential equation made visible. A solution is then a curve that is tangent to the field everywhere along it, which is a thing that can be traced with a pencil.

The three curves in the hero figure are traced numerically, one small step at a time, from the field alone. That is a deliberate choice: a figure that plotted $y_0e^{kx}$ and drew a field around it would be showing that a known answer fits, whereas walking the field is solving the equation, and the closed form arriving at the end of it is the result rather than the starting point.

There is a reason to insist on that. Every one of the numbers printed under these figures — the doubling interval, the agreement with the closed form, the ratio measured over two intervals — is a measurement of the traced curve. Had the curve been plotted from $y_0e^{kx}$, the agreement would be an identity and the assertion would be checking that a formula equals itself. The figures are set up so that the check could fail, which is the only reason its passing means anything.

Existence is easy and uniqueness is the point #

That some solution exists is not surprising — the field is smooth and a curve can be traced along it. What has to be established is that there is only one curve through each starting point, because that is the claim doing all the work whenever an exponential is used to describe anything.

The argument is one line and it is a small marvel. Suppose $y$ satisfies $y' = ky$. Consider the auxiliary function

$$g(x) = y(x)\,e^{-kx}.$$

Differentiating with the product rule, $g' = y'e^{-kx} - ky e^{-kx} = (y' - ky)e^{-kx} = 0$. A function with zero derivative everywhere is constant, so $g(x) = g(0) = y(0)$, and therefore

$$y(x) = y(0)\,e^{kx}.$$

Every solution is of that form. Not a solution has that form — every one.

The trick is worth staring at: the equation was multiplied by whatever it took to make the left-hand side an exact derivative, and then the only fact used was that a function whose derivative vanishes cannot go anywhere. That last fact is a consequence of the mean value theorem and is the whole reason this uniqueness argument works, which is a good example of a theorem whose importance is entirely in what it lets other arguments assume.

Why the doubling time does not depend on the doubling #

The consequence that gets used most is this one. If the amount doubles over some interval, it doubles over every interval of that length, wherever the interval is placed.

The reason is the multiplicative form of the solution: $y(x + T)/y(x) = e^{kT}$, in which $x$ has cancelled out. Setting that ratio to 2 gives the doubling time

$$T = \frac{\ln 2}{k},$$

which depends on the rate and on nothing else.

This is not a small observation dressed up. It is the property that makes a half-life a meaningful number. A quantity of radioactive material has a half-life because the decay rate is proportional to the amount present; a quantity draining through a hole in a bucket does not, because the rate depends on the pressure and therefore on the height, which is not proportional to the volume for any bucket that is not a cylinder. Half-life is not a general concept about decaying things. It is a property of one differential equation.

That is worth stating as a fact about the family. There is one equation for each $k$, and one curve shape for each base, and the correspondence between them is $b = e^k$. So the apparent variety of exponential curves is a single curve read at different speeds, which is why choosing a base other than $e$ costs a conversion factor and never anything more.

The limit definition, arriving from the equation #

The equation also explains where the familiar limit comes from.

Suppose the growth is applied not continuously but in $n$ equal instalments over one unit of time, each adding a fraction $k/n$ of the current amount. After $n$ steps the total factor is $(1 + k/n)^n$. As the instalments are made finer this is exactly the process of tracing the slope field in steps of $1/n$ — Euler’s method, the crudest possible way of following the field — and the limit is the solution at $x = 1$:

$$\lim_{n\to\infty}\left(1 + \frac{k}{n}\right)^n = e^k.$$

So the compound-interest definition is not a separate route to $e$. It is the numerical solution of the differential equation, with the step size going to zero, and it converges for the same reason any decent numerical method does.

What the field is not #

A slope field is a useful picture and it invites a misreading worth heading off.

It looks like a flow — as though something were being carried along the segments — and for this equation that reading happens to be harmless, because $x$ is the independent variable and time is a fair name for it. For most differential equations it is wrong. A slope field is a picture of a constraint on graphs, and the segments say only that a solution passing through here must be heading that way. Nothing is moving.

The distinction matters as soon as the field’s slope depends on $x$ as well as $y$, because then the segments in one column differ from those in the next and a curve traced through them is not following anything; it is satisfying a condition point by point. The equation here is unusually cooperative — its field is the same in every column, which is exactly the statement that the doubling time does not depend on where the doubling starts — and it is worth knowing that the cooperativeness is a property of this equation and not of slope fields.

Where the equation is not quite right #

The honest reason to spend a rung on this is that the equation is used constantly and is nearly always an approximation, and the interesting part is where it breaks.

$y' = ky$ says the growth rate per individual is a constant, independent of how many there are. For a population that stops being true once the members compete for anything, and the standard repair adds a term: $y' = ky(1 - y/M)$, which grows exponentially while $y$ is small and levels off at $M$. That is the logistic equation, and iterating its discrete version is where the road paved with doublings starts — a reminder that the well-behaved equation here is a special case of something that is not well behaved at all.

For radioactive decay the equation is exact, because the decay of one nucleus genuinely does not depend on how many others there are, and it is exact only in the sense that a large number of independent events has a smooth average. With ten remaining atoms the smooth curve is a poor description of what happens, and what happens instead is a matter of counting.

For money the equation is a convention. Nothing continuous happens to a bank balance, and continuous compounding is a modelling choice that makes the arithmetic pleasant.

The pattern across the three is worth naming. The equation is exact where the mechanism genuinely has no memory and no interaction — one nucleus decaying regardless of the others — and it is an approximation everywhere else, good while the quantity is small enough that whatever it will eventually run into has not been reached. Exponential growth is therefore always a claim about a regime, and a curve fitted to data over a range says nothing whatever about what happens outside it. Every famous failure of an exponential extrapolation is a failure to notice that.

The same equation, elsewhere #

The reason to give this equation a rung of its own is how much of the rest of mathematics turns out to be it.

Newton’s law of cooling says the temperature difference from the surroundings obeys it. A capacitor discharging through a resistor obeys it. The concentration of a drug being cleared by a liver obeys it, which is why doses are prescribed on a schedule set by a half-life. Continuously compounded interest obeys it by definition.

And it appears in a form with no time in it at all. A contraction mapping reduces the distance to its fixed point by a constant factor each step, so the error after $n$ steps is $c^n$ times the error at the start — the discrete version of the same equation, and the reason the convergence is called geometric. So is the shrinking angle in the power method. The equation governs anything whose change is proportional to its state, and that description fits an unreasonable number of things.

What the sign decides, and nothing else #

Everything about the qualitative behaviour is decided by one number, and it is worth listing what is not decided by anything else.

If $k > 0$ every solution grows without bound in one direction and to nothing in the other. If $k < 0$ the same picture is reflected. If $k = 0$ nothing moves. There is no other case: no oscillation, no equilibrium other than the origin, no dependence on the starting height beyond a scale factor. Two solutions with different starting heights are the same curve, one stretched vertically — since $y_0e^{kx}$ is $y_0$ times a fixed function — so the whole family is one shape.

That is unusually poor as behaviour goes, and it is why the equation is the base case rather than an interesting one. The moment two such quantities are allowed to feed into each other, the possibilities open up completely: growth, decay, spirals, and closed orbits all become available, and which one occurs is decided by the eigenvalues of a two-by-two matrix. That is the next rung.

What the picture cannot show #

The field is a picture of the equation and the curves are pictures of solutions, and the one thing that cannot be drawn is the uniqueness. Three curves are shown; the claim is about all of them, including the ones through every point not drawn. A picture of a field is compatible with a solution that branches, or with two solutions crossing, and nothing about the visual density of a slope field rules those out.

They are ruled out by the one-line argument above, and that argument is doing something the picture cannot even gesture at: it considers an arbitrary solution about which nothing is known and shows it must be the familiar one. There is no drawing of an arbitrary solution.

The field also flatters the equation. Every segment in it is short, and a short segment near a solution curve always looks tangent to it. The gap between “the field points that way here” and “a curve following the field arrives there” is exactly the numerical error the figure measures and controls, and a slope field drawn coarsely enough to read is drawn too coarsely to show it.

The one-line argument, and why it generalises #

The uniqueness proof multiplied by $e^{-kx}$ and got a derivative that vanished. That factor is called an integrating factor, and choosing it is the whole method for a large class of equations.

For $y' + p(x)y = q(x)$, multiplying by $\mu(x) = e^{\int p}$ makes the left-hand side exactly $(\mu y)'$, after which both sides can be integrated and the answer read off. The case here is $p$ constant and $q$ zero, which is why the factor was a plain exponential and the right-hand side was nothing at all.

What is worth noticing is the shape of the trick rather than the formula. A quantity that was not obviously conserved — $y$ itself is changing all the time — was multiplied by something chosen so that the product is conserved, and then the conservation did the work. That is the same move as finding a quantity a process cannot change and reading the answer off it, which turns up in a signed count that a deformation cannot alter and in a parity that no redrawing can change. Here the invariant had to be manufactured, which is the harder and more useful version.

The ladder from here #

Rungs above: the exponential of a matrix, where the same equation is solved with a square array in the place of $k$ and several quantities grow into each other. Where $e$ turns up in counting, with no rate of change anywhere. Second-order equations, where two exponentials combine and can become an oscillation. Separation of variables, of which the argument here is the simplest case. Systems of equations and their eigenvalues, which decide whether the whole thing grows, decays or spirals. And the equations that have no closed-form solution at all, where the slope field is not an illustration but the only description available.