ODE-33

Laplace of e^{-at}

Transform of a decaying exponential.

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GoverningLaplace of e^{-at}

Governing equation

1/(s+a)

$s \in \left[0.1, 12\right]$

where

s: s (1/s)
a: a (1/s)
Y: Laplace of e^{-at} (s)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

1/(s+a)

$s \in \left[0.1, 12\right]$

$s = 0.1$ $Y = 1.667$

Along the curve

②

Domain of the variable — blue

s_{\min} = 0.1,\quad s_{\max} = 12

$s \in \left[0.1, 12\right]$

$s = 0.1$ $s = 0.1$

Along the curve

③

Response curve — green

1/(s+a)

$s \in \left[0.1, 12\right]$

$s = 0.1$ $Y = 1.667$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}Y}{\mathrm{d}s}

$s \in \left[0.1, 12\right]$

$s = 0.175$ $Y^{\prime} = -2.223$

Along the curve

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-33 — Laplace of e^{-at}) is the form associated with Laplace of e^{-at}. Working symbols: $s$ , $a$ $\rightarrow$ $Y$ . Transform of a decaying exponential. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

Y

from

s

a

in Differential equations via

1/(s+a)

Transform of a decaying exponential. 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

s = 2.000\,\mathrm{1/s}

a = 0.500\,\mathrm{1/s}

, the governing relation

1/(s+a)

yields

Y = 0.400\,\mathrm{s}

. 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

s s2 1/sa a0.5 1/s

Outputs

Laplace of e^{-at} Y0.400 s

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Narration of this film

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

Transform of a decaying exponential. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube