Differential Equations

Every differential equation in this Subject is a sentence about a rate, so the rate has to be something you can compute and not only something you can recite. Here it is computed first and named second. The average rate of change over [a, a + h] is a fraction, and the lab prints it as one for a column of halving steps; the derivative is the number that column heads for, stated as a claim the table illustrates and does not prove. For a polynomial the claim is not even needed, because the quotient simplifies to a polynomial in h and the derivative is its constant term. The power, product and chain rules are then checked exactly on polynomials by expanding both sides, and the course ends on the exponential, the one place where a quotient cannot be exact and the lab says so by rounding and labelling every figure.

An integral is accumulated change. This course adds up a rate piece by piece in exact fractions, watches the sums close in on a number, and then shows that reversing the power rule finds that number without any sum. The one constant that reversal leaves open is the reason an equation later needs an initial value.

A differential equation does not hand over a number or a formula. It states how a quantity changes, and the first job is to learn what counts as an answer: a function, usually a whole family of them, which is accepted or rejected by one substitution and one residual that either is zero for every t or is not. The second job is to see the equation without solving it, as a field of short segments, and to read off what the field settles and what it only suggests. The third is to compute. Euler's method is the sentence “the next value is this value plus the rate times the step”, applied in exact fractions, and the error against a known solution is exact too, so the course can show — with the ratio of two errors, not with a claim — that halving the step halves Euler's error, quarters the improved method's, and divides Runge–Kutta's by sixteen. It closes on the place all of this fails: a solution that ceases to exist, which a method keeps stepping past without noticing.

One method carries this whole course: write the equation as g(y)·y′ = h(t), find an antiderivative of each side, and the relation between y and t falls out with one constant in it. What the course adds to the method is care about what it produces. An answer such as y² = t² + 4 is not yet a function; it becomes one only when an initial value picks a sign, and it stops being one where the square root runs out. The simplest equation of all, y′ = k·y, gets the most attention, because Euler's method on it is an exact geometric sequence the lab prints beside e^(kt), and the gap between the two is the same gap the previous course measured. Every figure that is a fraction is printed as one; every value of e^(kt) and every ln 2 is rounded to six figures and marked ≈, and the lessons say so wherever one appears.

The last course solved equations: it separated the variables and integrated. This one asks a different question of the same first-order equations, and it is often the more useful one. For y′ = f(y) nobody needs a formula to say that a solution starting at 2 settles at 1, or climbs without limit, or that a small change in a parameter makes two equilibria collide and disappear. All of it comes from the sign of f, which a browser can evaluate exactly, and from one derivative. The lab keeps the exact side exact: equilibria are fractions or surds, the slope at each is an exact number, and the only things that are drawn rather than computed are the solution curves, which the legend says are stepped in floating point. The method is approximate, the classification is not.

One equation does a great deal of work. A tank filling, a capacitor charging, a loan being repaid and a cooling cup of coffee are all y′ + a·y = b with different letters, and every one of them is a constant plus a gap that shrinks by a fixed factor in equal times. This course solves that equation three ways: as a shifted exponential, by multiplying through by an integrating factor that makes the left side a derivative, and as a homogeneous solution plus one particular solution found by a guess whose coefficients are matched exactly. The cases that are exact throughout are the ones with p constant or a/t, where the factor is an exponential or a power of t. The last lesson turns to the method rather than the equation: Euler's factor 1 − a·h is an exact fraction, and it leaves the interval from −1 to 1 at a step size you can compute.

A mass on a spring, a pendulum at small angles and a circuit with an inductor all obey the same equation: the rate of the rate of the unknown, a multiple of its rate, and a multiple of the unknown itself add up to zero. Separable Equations, Growth and Decay solved y′ = k·y with e^(kt); this course tries the same exponential on the second-order equation and finds that it works, provided r is a root of a quadratic. The quadratic is the one from Algebra, and its three possible discriminants give three shapes of solution: two exponentials, an exponential with a factor of t, and an exponential multiplied by a cosine and a sine. Every closed form is checked by substituting it into the equation, and the two constants are fitted from the starting value and the starting rate. The last two lessons look at the same equation from outside, as a pair of first-order equations whose matrix carries the quadratic inside its trace and determinant, and show Euler's method pushing a circle's worth of motion steadily outward.

The previous course solved a·y″ + b·y′ + c·y = 0 as a piece of algebra: find the roots of one quadratic, and the roots decide the shape. This course gives the letters their meanings. m is a mass, k a spring, c a drag, and the roots of the quadratic become frequencies and decay rates you can name and compare. The questions change with the meaning: how fast does it swing, how big, how long until it is still, and what happens when something keeps pushing. The lab for the whole course is one oscillator whose exact quantities appear as tiles — ω₀², c² − 4mk, the critical damping, the amplitude squared, the forced response — with the motion itself drawn by stepping in floating point and labelled as a drawing. A figure that involves π, a square root or an exponential is rounded to six figures and marked ≈; everything else is a fraction.

A single equation gave a curve against time. A pair of equations gives a curve in a plane, and the plane has no time axis: time is the direction you travel along the curve. For a linear system x′ = a·x + b·y, y′ = c·x + d·y the whole picture follows from a two-by-two matrix. Its eigenvalues solve one quadratic, λ² − τ·λ + Δ = 0, whose coefficients are the trace and the determinant, so two exact numbers say whether solutions rush in, rush out, circle or do both. The labs keep that side exact: traces, determinants, eigenvalues, eigenvectors and fitted constants are fractions or surds. What is drawn rather than computed is the curve itself, which the legend says is stepped in floating point. The second half asks what survives when the system is not linear, and the honest answer is: near an equilibrium, most of it.

Every differential equation so far has been solved in the time domain: guess a form, match coefficients, then fit the constants to the starting values. The Laplace transform moves the whole problem to a different variable, s, where y′ becomes s·Y − y(0) and y″ becomes s²·Y − s·y(0) − y′(0). The equation turns into (a·s² + b·s + c)·Y = something, a line of algebra, and the starting values are already inside it. What is left is a rational function to take apart into the pieces of a short table. This course builds the table, proves the derivative rule from the product rule, solves initial value problems end to end, and then reads what the poles of the answer say about the long run. Every transform and every coefficient in the labs is an exact fraction, and every answer is checked by substituting it back.