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.
Overview
By the end you can recognise a separable equation, integrate both sides exactly, and say what the answer is and where it holds. You can then use the one equation y′ = k·(y − A) for growth, decay, dating, cooling and mixing, and read a bounded growth law and a harvested one from their right-hand sides.
Recognise g(y)·y′ = h(t), integrate both sides exactly, and write the implicit solution with its constant taken from an initial value — checking each antiderivative against the lab rather than trusting a sign.
Explain separation as the chain rule read backwards, and verify a solution by differentiating the implicit relation to get the equation back.
Take the branch the initial value demands, and state the interval on which the explicit solution exists — including the one that ends in finite time.
Solve y′ = k·y by separation, get Euler's yₙ = y₀·(1 + kh)ⁿ exactly, and read the gap between that sequence and the rounded e^(kt) at the same time.
Derive T = ln 2/k from e^(kT) = 2, state it as rounded, and find the first step at which the exact Euler sequence passes double or half.
Write cooling, mixing and logistic growth as first-order equations, find their equilibria, and compute the harvest threshold r·K/4 above which a population collapses.
Lessons
The method first, in three lessons: how to separate and integrate, why that is legitimate, and what the answer means once it is an implicit relation. Then the exponential equation and its two readings, a rate and a doubling time. Decay, dating, cooling and mixing follow as one equation with a shift in it. The last part adds a bound to the growth, and then removes some of it again.
Differential Equations and Euler's Method, which supplies what a solution is, how an initial value fixes one, and Euler's steps as exact fractions, and through it Accumulation and the Integral for antiderivatives and the constant of integration. From Algebra, Exponential and Logarithmic Functions for e, the logarithm and why e^(kt) doubles in a fixed time, and the geometric sequence from Sequences and Series, which Euler's method on a growth equation turns out to be.
Write the equation in the form g(y)·y′ = h(t) before you integrate anything. If it will not go into that form, the method does not apply, and no amount of cleverness with the integral signs will make it. The lab refuses a g it cannot integrate and says which one.
Check every solution by differentiating it. The lab does this with the chain rule and prints equal or not, and the habit is worth more than the printout: a sign lost in an antiderivative produces a perfectly smooth wrong answer that no later step will flag.
Say which tier a number is in. The Euler sequence is a column of exact fractions, e^(kt) and ln 2 are rounded and carry ≈, and the difference between them is the method's error at that step size and no other. Quote the step size with the figure.
Treat a model as an equation someone gave you. Every application here states its assumptions — a well-stirred tank, a constant rate constant, a population with a fixed carrying capacity — and the lab computes what follows from them; whether the assumptions hold for a real tank is not something it can tell you.
Equations that cannot be separated. y′ = t + y has no g(y)·y′ = h(t) form, and the method of the first-order linear course handles it instead. Exact equations, Bernoulli equations and substitutions that turn an equation into a separable one are techniques for finding closed forms the exact lab cannot verify, and they are left out on purpose.
Integrals beyond polynomials and the two reciprocals. The lab separates g(y) that is a polynomial in y, or exactly 1/y or 1/y^2, and h(t) that is a polynomial in t. Partial fractions in y, which the full logistic solution needs, are not built here: the logistic lessons read the equation from its equilibria and from exact Euler steps instead of from a formula.
Existence and uniqueness. A separable equation can have a solution that stops in finite time, and the course shows one. Why an initial value picks exactly one solution, and when it does not, is stated in words in the previous course and not proved anywhere.
Modelling as a skill. The tank, the cup of coffee, the carbon sample and the population are given with their equations. Deriving such a model from measurements, fitting its constants and judging whether it fits are a statistics course's work.