ODE-31

Characteristic root

Real part proxy of a quadratic characteristic.

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GoverningCharacteristic root

Governing equation

r=(-b\\pm\\sqrt{b^2-4ac})/(2a)

where

a: a (—)
b: b (—)
c: c (—)
rp: Characteristic root (—)

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-31 — Characteristic root) is the form associated with Characteristic root. Working symbols: $a$ , $b$ , $c$ $\rightarrow$ $rp$ . Real part proxy of a quadratic characteristic. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

rp

from

a

b

c

in Differential equations via

r=(-b\\pm\\sqrt{b^2-4ac})/(2a)

Real part proxy of a quadratic characteristic. 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{—}

b = 3.000\,\mathrm{—}

c = 2.000\,\mathrm{—}

, the governing relation

r=(-b\\pm\\sqrt{b^2-4ac})/(2a)

yields

rp = -1.000\,\mathrm{—}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a1 —b b3 —c c2 —

Outputs

Characteristic root rp-1.000 —

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Real part proxy of a quadratic characteristic. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube