ODE-01

Exponential solution

Linear 1st order.

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ODEy'=ky

Governing equation

y=y_0 e^{kt}

where

y_0: y0 (—)
k: k (1/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-01 — Exponential solution) is the form associated with y'=ky. Working symbols: $y_0$ , $k$ , $t$ $\rightarrow$ $y$ . Linear 1st order.

Purpose

Purpose: compute

y

from

y_0

k

t

in Differential equations via

y=y_0 e^{kt}

Linear 1st order. 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

y_0 = 1.000\,\mathrm{—}

k = -0.300\,\mathrm{1/s}

t = 2.000\,\mathrm{s}

, the governing relation

y=y_0 e^{kt}

yields

y = 0.5488\,\mathrm{—}

. Linear 1st order. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

y0 y_01 —k k-0.3 1/st t2 s

Outputs

y y0.5488 —

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Narration of this film

Linear 1st order.

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