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

Maximum Sustainable Harvest: Why h = a^2/4b

Why do the two critical points of $ay - by^2 - h$ collide at $h = a^2/(4b)$, and what happens to the fish once $h$ is larger?

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Your summary note

1. Lowering the parabola $ay - by^2 - h$
  Draw the graph for $h = 0$ and two larger values instead of using the quadratic formula, and record that it drops by $h$ with its shape unchanged while the two roots move inward.
2. The critical harvest rate $h_m = \frac{a^2}{4b}$
  Locate the peak of $ay - by^2$ at $y = \frac{a}{2b}$ , compute its height, record the two critical points merging there and vanishing for $h > h_m$ , and confirm with $a^2 - 4bh = 0$ .
3. Solution curves in the three harvesting regimes
  Sketch the solutions for $h < h_m$ , $h = h_m$ and $h > h_m$ , and classify the critical points in each: an unstable lower and stable upper pair, one semi-stable point at $\frac{a}{2b}$ , or none.
4. What $h_m$ means for the salmon farm
  State $h_m$ as the greatest harvest rate that avoids extinction, and record the fate of a population started above the lower critical point under a harvest at most $h_m$ and one just above it.
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