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 Boat's Path: A Logarithmic Spiral in Polar Coordinates

What curve does the escaping boat actually trace, and why does its solution only make sense once you switch to polar coordinates?

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Your summary note

1. Separating the boat equation with $y = zx$
  Substitute $y = zx$ to reach $x\frac{dz}{dx} = \frac{z+1}{1-z} - z$ , combine the right side over $1 - z$ into $\frac{1+z^2}{1-z}$ before inverting, and separate as $\frac{1-z}{1+z^2}\,dz = \frac{dx}{x}$ .
2. Integrating $\frac{1-z}{1+z^2}$ term by term
  Integrate $\frac{1}{1+z^2}$ to $\arctan z$ and $\frac{z}{1+z^2}$ to a logarithm with the right constant factor, reaching $\arctan z - \frac{1}{2}\ln(1+z^2) = \ln x + c$ .
3. Returning to $x$ and $y$
  Apply the laws of logarithms, substitute $z = y/x$ , and bring $x$ inside the square root (for $x > 0$ ) to reach $\arctan\frac{y}{x} = \ln\sqrt{x^2+y^2} + c$ .
4. Polar coordinates and $r = Ce^{\theta}$
  Record that the Cartesian answer is solved technically but not morally, identify $\theta$ and $r$ in it, write $\theta = \ln r + c$ , and exponentiate to the exponential spiral $r = Ce^{\theta}$ .
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