ODE-30

Picard first iterate

Mean-value first Picard.

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GoverningPicard first iterate

Governing equation

y_1=y_0+\\int f

$h \in \left[0.05, 2\right]$

where

y0: y0 (—)
favg: favg (—)
h: h (—)
y1: Picard first iterate (—)

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

y_1=y_0+\\int f

$h \in \left[0.05, 2\right]$

$h = 0.05$ $y1 = 1.02$

Along the curve

②

Domain of the variable — blue

h_{\min} = 0.05,\quad h_{\max} = 2

$h \in \left[0.05, 2\right]$

$h = 0.05$ $h = 0.05$

Along the curve

③

Response curve — green

y_1=y_0+\\int f

$h \in \left[0.05, 2\right]$

$h = 0.05$ $y1 = 1.02$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}y1}{\mathrm{d}h}

$h \in \left[0.05, 2\right]$

$h = 0.062$ $y1^{\prime} = 0.4$

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-30 — Picard first iterate) is the form associated with Picard first iterate. Working symbols: $y0$ , $favg$ , $h$ $\rightarrow$ $y1$ . Mean-value first Picard. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

y1

from

y0

favg

h

in Differential equations via

y_1=y_0+\\int f

Mean-value first Picard. 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

y0 = 1.000\,\mathrm{—}

favg = 0.400\,\mathrm{—}

h = 0.500\,\mathrm{—}

, the governing relation

y_1=y_0+\\int f

yields

y1 = 1.200\,\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

y0 y01 —favg favg0.4 —h h0.5 —

Outputs

Picard first iterate y11.200 —

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Narration of this film

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

Mean-value first Picard. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube