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.
Overview
By the end you can say what a differential equation asks for, decide whether a proposed function answers it, draw and read its slope field, and carry out each of the three step methods by hand and in the lab, with the error stated and the failure of the method named.
Differentiate the candidate, put it in the equation, simplify the residual symbolically and decide: zero for every t is a solution, anything else is not, and a function that satisfies the equation at one value of t has shown nothing. Equations are named by order and sorted as linear or not.
State how many initial values an equation needs — one for a first-order equation, two for a second-order one — solve for the constants in a family of solutions, and check that the result still has residual zero.
Compute the slope at a grid point exactly, draw the segment, find the curve where the slope is zero, and say which conclusions the field settles (horizontal solutions, the direction of travel, that curves cannot cross) and which it only suggests.
Carry out Euler's method by hand for several steps, read the table, and say what the method assumes inside each step; the fractions are right and the answer is still only as close as the step allows.
Compute the error at three step sizes, form the exact ratios and read the order: Euler's method is first order, the improved method second, Runge–Kutta fourth, each shown by a ratio that is exactly 2, 4 or 16 on a chosen equation.
Show that y = 1/(1 − t) solves y′ = y² and ends at t = 1, watch Euler's method step past it, and read the digit budget that stops an exact computation whose fractions double in length.
Lessons
The object first: what an equation about a rate is, how a proposed solution is tested, and how an initial value picks one curve from the family. Then the equation as a picture, a slope field read without solving anything. Then the computing, in order of cost and of accuracy — Euler's method, its error against the step size, the improved method and Runge–Kutta — and last the equation whose solution does not last, which is where a method that only ever looks one step ahead is least trustworthy.
Accumulation and the integral, and through it rates of change and the derivative: the difference quotient and what it approaches, the derivative of a polynomial and of 1/t, the product and chain rules on polynomials, the antiderivative and the constant that comes with it, and a sum of rate times step as total change. The constant of integration is the whole reason an initial value is needed here, and the left sum is exactly Euler's method on the simplest equation, so both are leaned on: the constant from the first lesson, the left sum from “Euler's Method” on. From algebra, the exponential function and the number e, and the habit of checking an answer by substituting it back.
Check before you trust. Every closed form on this course is tested by substituting it and reading the residual, and the lab does that symbolically, so the verdict is a fact about every t and not about the five you happened to try. Make the same habit yours: a candidate that works at one value of t, or for one choice of the constant, has not been checked.
Say which tier a number is in. A fraction such as 625/256 is exactly what Euler's method produces and is exactly (5/4)⁴; the value of the true solution is often irrational and is printed as approximate, to six figures, and labelled rounded. Keep the two apart in your own working, because the gap between them is the error the course measures.
Treat the method as a recipe and not as the answer. A step method assumes the rate stays what it was at the start of the step; nothing computed from it is the solution, only a value near it for a reason the error lessons can quantify. When a lab reports an error it is the error for that step size alone.
Read the pictures for what they settle. A slope field gives the slope at each point exactly and the long-run behaviour only suggestively. Write down which of your conclusions came from the sign of the slope and which from the look of the curve, and trust them accordingly.
Existence and uniqueness proofs. “Blow-Up and the Interval of Existence” states in words what makes a solution through a given point exist and be the only one, and shows the equation that breaks the global version; the proof, the Lipschitz condition and the examples with more than one solution through a point are a course in analysis and are not here.
Numerical analysis as a subject. Adaptive step size, error estimators, multistep methods, implicit Runge–Kutta and the theory of A-stability are all absent. The course shows the order of a method by exact ratios on chosen equations and stops; the one exact factor that explains why a method fails on a decaying equation comes in First-Order Linear Equations.
Floating-point stepping as a computed result. Where an equation has no closed form the slope fields draw a curve by stepping in floating point, and the legend says it is drawn; nothing in a tile comes from it. A reader who wants a nonlinear equation's numbers is told the honest truth: stepping a quadratic right-hand side in exact fractions reaches the digit budget in about eleven steps, and floating point hides that by rounding at every step.
Methods for finding closed forms of general equations. Exact equations, Bernoulli and Riccati equations and substitutions are not here; the equations whose solutions can be checked exactly are the ones the course uses, and the next courses teach the families that can actually be solved by hand.
The claims this course states and does not prove. That the errors of the three methods shrink at the orders shown, which the exact ratios demonstrate on chosen equations and do not establish for every equation, and that a smooth right-hand side gives exactly one solution through a point near the start. Every figure printed is exact or labelled; neither claim is load-bearing for a number on the page.