Differential Equations

The Geometric View of ODEs

01Direction Fields and the Geometric View of ODEs 02Integral Curves and the Isocline Method 03Perpendicular Isoclines: The Circles of y' = -x/y 04Isocline Corridors and a Trap for Solutions 05Existence and Uniqueness: One Curve Through a Point 06Where Existence and Uniqueness Fail Problem set0/10 Problem set 20/10 MIT problem set0/3

Numerical Methods

01Euler's Method and Its Three Equations 02Euler's Method by Hand: Building the Table 03Convexity and the Sign of Euler's Error 04First Order Error: Halve the Step, Halve the Error 05Improved Euler (RK2): Averaging Two Slopes 06Runge-Kutta 4 and the Price of Accuracy 07Singular Points: Where Every Method Fails Problem set0/10 Problem set 20/10 MIT problem set0/1

First-Order Linear Equations

01First Order Linear: Standard Form and the Sign of p 02The Conduction-Diffusion Model: Newton's Law of Cooling 03The Integrating Factor and the Four-Step Method 04Integrating Factors: Two Mistakes and a Trick Question 05Constant Coefficients: Steady State and Transient Problem set0/10 Problem set 20/10 MIT problem set0/1

Substitution Methods

01Scaling a First Order ODE: The Fourth-Power Cooling Law 02The Bernoulli Equation and the Substitution v = y^(1-n)03A Worked Bernoulli Example: The Substitution v = 1/y 04Homogeneous Equations: The Zoom Test and y = zx 05Modelling the Lighthouse Chase: A 45-Degree Heading 06The Boat's Path: A Logarithmic Spiral in Polar Coordinates Problem set0/10 Problem set 20/10

Autonomous Equations

01Autonomous Equations and Time-Translation Invariance 02Critical Points: Constant Solutions as Barriers 03The Embezzler Model dy/dt = ry - w: One Unstable Balance 04The Logistic Equation: Carrying Capacity Without Solving 05Stable, Unstable and Semi-Stable Critical Points 06Maximum Sustainable Harvest: Why h = a^2/4b Problem set0/10 Problem set 20/10 MIT problem set0/3

Complex Numbers

01Complex Division, Polar Form and the Definition of e^(iθ)02Why e^(iθ) Earns Its Name: Multiply Moduli, Add Angles 03Integrating e^(-x) cos x by Complexifying 04The nth Roots of Unity: Zeta on the Unit Circle Problem set0/10 Problem set 20/10 MIT problem set0/4

Input and Response

01Steady-State and Transient Solutions of y' + ky = q(t)02Input, Response and the Superposition Principle 03Solving y' + ky = k cos ωt by Complexification 04Amplitude and Phase Lag of a Sinusoidal Response Problem set0/10 Problem set 20/10 MIT problem set0/2

01Cartesian Method: Rationalise, Then Take the Real Part 02The Sinusoidal Identity and Its Right Triangle 03Three Proofs of the Sinusoidal Identity 04The Mixing Tank: Salt Balance and k = r/V 05RC Circuits, Decay Chains and Negative k Problem set0/10 Problem set 20/10 MIT problem set0/2

Second-Order Linear Equations

01Second-Order Linear ODEs and the Spring–Mass–Dashpot 02The Characteristic Equation and Overdamped Motion 03Complex Roots a ± bi and Underdamped Oscillation 04Critical Damping and Reduction of Order: t e^(−at)Problem set0/10 Problem set 20/10 MIT problem set0/2

The nth Roots of Unity: Zeta on the Unit Circle

Why does $z^5 = 1$ have five solutions, and why do they sit evenly spaced around the unit circle?

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Your summary note

1. Counting the solutions of $z^n = 1$
  State that $1$ has exactly $n$ complex $n$ th roots, and contrast this with the real case, recording how many real roots there are for odd $n$ and for even $n$ .
2. Equal spacing by the angle $\alpha = \frac{2\pi}{n}$
  Draw the unit circle for $n = 5$ with five equally spaced points starting at $1$ , mark the angle $\frac{2\pi}{5}$ , and record the multiplication rule for moduli and angles that places the roots there.
3. Verifying $\zeta = e^{2\pi i/5}$ algebraically
  Write $\zeta$ in polar form with modulus $1$ and angle $\frac{2\pi}{5}$ , then carry $\zeta^5 = e^{2\pi i} = 1$ through step by step using the exponential law, noting that angles $2\pi$ and $0$ coincide.
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