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

Complex Roots a ± bi and Underdamped Oscillation

How do the real and imaginary parts of one complex exponential give two real solutions, and why do complex roots mean a decaying oscillation?

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Your summary note

1. Real and imaginary parts of a complex solution are solutions
  State the theorem for $y'' + Ay' + By = 0$ with $A, B$ real, write out the proof by splitting into real and imaginary parts, and mark the step that needs $A, B$ real.
2. General solution $y = e^{at}(c_1\cos bt + c_2\sin bt)$
  Expand $e^{(a+bi)t}$ by Euler's formula to get $e^{(a+bi)t}$ and $y_2 = e^{at}\sin bt$ , then rewrite $c_1\cos bt + c_2\sin bt$ as $C\cos(bt - \phi)$ , $(C, \phi)$ being polar coordinates of $(c_1, c_2)$ .
3. Worked example $y'' + 4y' + 5y = 0$
  Solve the characteristic equation to $r = -2 \pm i$ , write the general solution, and apply $y(0) = 1$ , $y'(0) = 0$ to reach $y = e^{-2t}(\cos t + 2\sin t)$ .
4. The underdamped picture: oscillation inside a decaying envelope
  Convert the example to $\sqrt{5}\,e^{-2t}\cos(t - \phi)$ with $\tan\phi = 2$ , $\phi \approx 63^\circ$ , and sketch it between the envelopes $\pm\sqrt{5}\,e^{-2t}$ , marking the first zero at $\pi/2 + \phi$ .
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