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

01One Complex Root Is Enough: Conjugates and Real Solutions 02When a Damped Spring Oscillates: p Below ω₀ 03Damped Oscillations: Pseudo-Period and ω₁² + p² = ω₀² Problem set0/10 Problem set 20/10 MIT problem set0/1

01Superposition: Why Linearity Makes c₁y₁ + c₂y₂ a Solution 02The Wronskian and Abel's Theorem: Fitting Any Initial Values 03Normalized Solutions: Why cosh x and sinh x Solve y'' − y = 0 04Uniqueness: Why c₁y₁ + c₂y₂ Is Every Solution Problem set0/10 Problem set 20/10

Normalized Solutions: Why cosh x and sinh x Solve y'' − y = 0

Why is $\cosh x$, not $e^x$, the natural partner for $y'' - y = 0$ once you choose solutions by their starting values?

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Your summary note

1. Any independent pair gives the same family
  Write $u_1$ , $u_2$ as combinations of $y_1$ , $y_2$ , and show that the combinations of the $u$ 's and of the $y$ 's are the same collection of functions.
2. The normalized solutions $Y_1$ , $Y_2$ at $0$
  Write the defining conditions $Y_1(0) = 1$ , $Y_1'(0) = 0$ , $Y_2(0) = 0$ , $Y_2'(0) = 1$ , and check that $\cos x$ , $\sin x$ satisfy them for $y'' + y = 0$ .
3. Deriving $\cosh x$ and $\sinh x$ from $e^{x}$ , $e^{-x}$
  For $y'' - y = 0$ , solve the two $2 \times 2$ systems for $c_1$ , $c_2$ to get $Y_1 = \frac{e^x + e^{-x}}{2}$ and $Y_2 = \frac{e^x - e^{-x}}{2}$ , then verify their initial values.
4. Comparing the standard and normalized pairs
  Record which initial value rules out $e^x$ and $e^{-x}$ as $Y_1$ or $Y_2$ , and write down $(\cosh x)' = \sinh x$ and $(\sinh x)' = \cosh x$ .
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