🌀 Differential Equations

What you’ll learn

1. A Sentence About ChangeRecognise a differential equation as a constraint on a function's behaviour rather than a statement of its value, and fix the four words of vocabulary the rest of the course uses.F = ma is a differential equation: it says how the rate of change of the rate of change of position relates to force, never what position is. The unknown is a function, checked by differentiating rather than substituting — which is why dy/dx = 2y is solved not by e^(2x) alone but by every multiple of it. Order tells you how many conditions will pin down one solution; linearity marks the border of what can be solved in closed form.
2. The Sentence Nature Keeps ReusingEstablish the course's through-line: dy/dt = ky describes money, bacteria, radioactive decay and cooling coffee because all four share one structure — independence at the unit level.'Rate proportional to amount' is what you get whenever each unit of a quantity acts independently and identically: each dollar earns without consulting the others, each nucleus decays indifferent to its neighbours. The equation is therefore a fact about structure, not about any particular science, which is why one piece of work buys compound interest, bacterial growth, radiocarbon dating, Newton's cooling, drug clearance and capacitor discharge at once. The sign of k separates a fortune from a corpse.
3. Separating the VariablesSolve dy/dt = ky honestly by separation, and see both what the method buys and what it silently deletes.Getting the y's on one side and the t's on the other and integrating gives y = Ae^(kt) — every solution of the sentence is an exponential. The method does not care about linearity, so it also handles dy/dt = y², whose solution reaches infinity at a finite time rather than merely growing fast. Dividing by f(y) quietly deletes the constant solutions where f(y) = 0, and those are exactly the equilibria the next chapters are built on.
4. A Family of Answers, Until You Fix One PointUnderstand why a differential equation's solution is a family of functions, and why an nth-order equation needs exactly n conditions.'This grows at 5% a year' cannot tell you how much of it there is: rate is indifferent to scale, so the equation cannot distinguish a hundred dollars from a million. The arbitrary constant is the equation being honest about what it does not contain, and an initial condition supplies it. The count is physics rather than bookkeeping — knowing every force on a ball still leaves you unable to say where it is until you know its starting position and velocity.
5. Reading the Answer Off the EquationExtract complete qualitative behaviour — equilibria, stability, long-run fate — from an equation without solving it, and see where the uniqueness guarantee fails.dy/dt = f(t, y) hands you the slope at every point of the plane for free, so the slope field can be drawn without any solving and the solutions are the curves that follow it. Equilibria are found by setting the right-hand side to zero — pure algebra — and their stability by a sign check on either side. The existence and uniqueness theorem carries real fine print: y′ = y² stops existing at a finite time, and y′ = y^(1/3) from zero has more than one solution.
6. Growth That Notices the CeilingAdd one factor to the exponential model and understand why early data cannot distinguish the two, using equilibria rather than a formula to get the full behaviour.dP/dt = rP(1 − P/K) is the exponential equation when P is small — which is exactly the trap, since every exponential fit to early data is also a perfect fit to a logistic curve while the forecasts diverge by orders of magnitude. Setting the right-hand side to zero gives equilibria at P = 0 (unstable) and P = K (stable), and two sign checks give the entire qualitative story. The exact solution exists and adds essentially nothing: understanding an equation and solving it are different activities.
7. The Equation That Runs the WorldIntroduce m·x″ + b·x′ + k·x = F(t) and read each of its terms as a distinct physical argument.The spring term k·x is proportional to displacement and supplies the desire to return; the damping term b·x′ is proportional to velocity and is the only term that removes energy; the inertia term m·x″ is why the system overshoots and therefore why it oscillates at all. Oscillation is the argument between the spring and the mass, with the damper as referee. F(t) is anything pushing from outside, and setting it to zero gives a system left alone.
8. Guess an Exponential, and the Algebra Falls OutSolve the second-order linear equation via the characteristic polynomial and understand the three damping regimes as consequences of one discriminant.Guessing x = e^(rt) turns every derivative into a multiplication, and the shared factor divides out to leave m·r² + b·r + k = 0 — calculus becomes algebra, because the exponential is the one function whose derivative is a multiple of itself. The discriminant b² − 4mk sorts the world: overdamped creeps back, critically damped returns fastest without overshoot, underdamped rings inside a decaying envelope. Critical damping is a design target for shock absorbers and door closers precisely because both alternatives are worse.
9. The Circuit Is the SpringDeliver the course's payoff: the series RLC circuit is not analogous to the mass–spring system but is the identical equation, so every result transfers for free.Kirchhoff's voltage law on an L-R-C loop gives L·q″ + R·q′ + q/C = V(t) — the same object as m·x″ + b·x′ + k·x = F(t), with L for m, R for b and 1/C for k. Therefore the discriminant b² − 4mk becomes R² − 4L/C and still sorts the three cases; √(k/m) becomes √(1/LC) and is still what the thing rings at. What such systems share is a structure — energy stored in two forms, traded back and forth, and leaking — which is common enough to make this one equation extraordinarily valuable.
10. Push It, and Ask What Comes BackUnderstand forced oscillation, the transient/steady-state split, and resonance as both a tool and a hazard — attributing the disputed Tacoma Narrows mechanism honestly.Under F₀·cos(ωt) the response splits into a transient that decays and forgets, and a steady state at the driving frequency that persists. Its amplitude A(ω) = F₀/√((k − mω²)² + (bω)²) blows up near ω = √(k/m), where the first bracket vanishes and only damping resists. This selectivity is what a tuned radio circuit is for; the collapse of the Tacoma Narrows Bridge is often taught as its hazardous twin, though Billah and Scanlan and others argue the mechanism was aeroelastic flutter rather than forced resonance, and the profession does not universally agree.
11. Two Equations, Two SpeciesMove from one equation to a coupled system, and see how a single product term makes the problem nonlinear and unsolvable in closed form while remaining fully understandable.Hares and lynx have no equations of their own because each population's rate depends on the other's current value, so both must be written at once. The H·L encounter term appears in both equations with opposite signs — one creature's loss is exactly the other's income — and it presumes free mixing, an idealisation worth keeping in view. That product also makes the system nonlinear, defeating every technique from the characteristic-polynomial chapter; the resulting cycles are emergent rather than regulated, and neutrally stable, which is a known weakness of the model.
12. The Phase PlaneAdopt the geometric turn: stop asking for formulas and ask what the set of all solutions looks like.Plotting velocity against position rather than each against time makes each state a point and each solution a trajectory, with time implicit in the motion. A closed loop is periodic behaviour and conservation of energy drawn; an inward spiral is energy leaving, since a loop that closed would have to return to the same energy. The three damping cases reappear as three shapes, and Poincaré's insight in the 1880s — that the shape of the flow is knowable even when no formula is coming — is the foundation of dynamical systems and chaos theory.
13. When There Is No FormulaTake the equation at its literal word numerically, understand Euler's error behaviour, and close the course's through-line.Euler's method walks a short step along the slope the equation states, then looks again — crude, since the slope changes across the step, but wrong in a predictable way, with error proportional to h. That poor exchange rate is why real solvers sample the slope several times per step, but the idea underlies every weather model and physics engine; a method nearly useless for two centuries became the foundation of computational science once arithmetic got free. The course closes where it opened: a small vocabulary of sentences is shared among an enormous number of unrelated things.

Questions this course answers

Grounded in trusted sources

• William E. Boyce, Richard C. DiPrima & Douglas B. Meade — Elementary Differential Equations and Boundary Value Problems (11th ed., 2017)
• Morris W. Hirsch, Stephen Smale & Robert L. Devaney — Differential Equations, Dynamical Systems, and an Introduction to Chaos (3rd ed., 2013)
• Steven H. Strogatz — Nonlinear Dynamics and Chaos (2nd ed., 2015)
• Gilbert Strang — Differential Equations and Linear Algebra (2014)
• K. Yusuf Billah & Robert H. Scanlan — "Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks", American Journal of Physics 59:2 (1991)
• Alfred J. Lotka — Elements of Physical Biology (1925); Vito Volterra — "Variazioni e fluttuazioni del numero d'individui in specie animali conviventi" (1926)
• All curves in this course's charts are computed by numerically integrating the models stated on the page (see the sub-caption of each chart); they are not measurements of any real population, circuit or vehicle.

Every Wunder lesson is built from real, reputable sources — never invented.

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