ODE-02

Harmonic oscillator

Undamped.

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ODESHO

Governing equation

y=A\\cos(\\omega t+\\phi)

where

A: A (—)
\omega: ω (rad/s)
t: t (s)
y: y (—)

Lecture brief

Historical brief

Exponential decay, the harmonic oscillator, SIR epidemics, Lotka–Volterra and Bessel are the first solvable ODEs of nature and populations. The lab is rate, period and phase portrait in closed form. This sheet (ODE-02 — Harmonic oscillator) is the form associated with SHO. Working symbols: $A$ , $\omega$ , $t$ $\rightarrow$ $y$ . Undamped.

Purpose

Purpose: compute

y

from

A

\omega

t

in Differential equations via

y=A\\cos(\\omega t+\\phi)

Undamped. Use it when a real differential equations question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

A = 1.000\,\mathrm{—}

\omega = 2.000\,\mathrm{rad/s}

t = 1.000\,\mathrm{s}

, the governing relation

y=A\\cos(\\omega t+\\phi)

yields

y = -0.4161\,\mathrm{—}

. Undamped. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

A A1 —ω \omega2 rad/st t1 s

Outputs

y y-0.4161 —

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Narration of this film

Undamped.

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