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

Damped Oscillations: Pseudo-Period and ω₁² + p² = ω₀²

Why does more damping slow a damped oscillation down, and how does the right triangle $\omega_1^2 + p^2 = \omega_0^2$ give its pseudo-period?

Loading…

←Previous When a Damped Spring Oscillates: p Below ω₀

Your summary note

1. The roots $-p \pm i\omega_1$ and the solution form
  Define $\omega_1^2 = \omega_0^2 - p^2$ , rewrite the roots as $-p \pm i\omega_1$ , and write the real solutions both as $e^{-pt}(c_1\cos\omega_1 t + c_2\sin\omega_1 t)$ and as $A e^{-pt}\cos(\omega_1 t - \phi)$ .
2. The pseudo-period $2\pi/\omega_1$
  Define pseudo-period and pseudo-frequency for a solution that is not truly periodic, and show from $\omega_1 t_1 - \phi = \pi/2$ that the next downward axis crossing is at $t_2 = t_1 + 2\pi/\omega_1$ .
3. The right triangle with legs $\omega_1$ , $p$ and hypotenuse $\omega_0$
  Draw the triangle for $\omega_1^2 + p^2 = \omega_0^2$ and record the effect of increasing the damping $p$ on $\omega_1$ and on the pseudo-period, for a fixed spring and mass.
4. What $p$ , $\omega_1$ , $A$ and $\phi$ each depend on
  Write $p = c/2m$ and $\omega_1$ in terms of $c$ , $k$ , $m$ , record that both come from the ODE alone, and record that $A$ and $\phi$ are set by the initial conditions.
Attempt 1 of 2