Foundations — ODE solvers — Euler's method (derivation, global error)

This page builds every symbol the parent note fires at you — $y^{\prime }$, $f(x,y)$, $h$, $x_n$, $y_n$, $O(h)$, $L$, $\xi$, $e^{L(x-x_0)}$ — starting from a smart 12-year-old who has never seen any of them. Read top to bottom; nothing appears before it is earned.

1. A function and its graph — the ground floor

Before anything else: a function is a machine. You feed it a number $x$, it returns a number, which we call $y$. We write $y=g(x)$: "$y$ equals $g$ of $x$."

• Plain words: $x$ is the input (how far right), $y$ is the output (how high).
• Picture: plot the point $(x,y)$ for every input. The dots join into a curve — the graph.
• Why we need it: the whole topic is about finding an unknown curve $y(x)$. We must be crystal-clear that $y$ is a height that changes as $x$ moves.

2. Slope — how steep, and which way

Pick two points on any straight line. Slope answers: for every 1 step right, how many steps up?

$\text{slope}=\frac{\text{rise}}{\text{run}}=\frac{\text{change in }y}{\text{change in }x}$

• Plain words: rise over run. Positive = uphill, negative = downhill, zero = flat.
• Picture: a right triangle hugging the line — horizontal leg is the run, vertical leg is the rise.
• Why we need it: Euler's whole trick is "walk in the direction of the slope." If you can't read a slope off a triangle, nothing later makes sense.

3. The derivative $y^{\prime }$ — the slope of a curve

A curve bends, so its steepness changes from point to point. The derivative is the slope of the curve at one single point — the slope of the tangent line just kissing the curve there.

We write it $y^{\prime }$ (read "y prime") or $\frac{dy}{dx}$ (read "dee-y by dee-x").

• Plain words: $y^{\prime }$ = the instantaneous steepness of the curve at a point.
• Picture: zoom into the curve until it looks straight; that straight line's slope is $y^{\prime }$.
• Why we need it: the differential equation is made of $y^{\prime }$. It is the star of the whole show.

4. The tangent line — the object Euler actually walks along

• Plain words: the tangent is the straight line touching the curve at one point with the same slope as the curve there.
• Picture (figure above): the curve in violet, its tangent in orange at one point. Notice the tangent leaves the curve as you move away — that gap is the seed of Euler's error.
• Why we need it: Euler replaces the curve, over one small step, by its tangent. The gap between tangent and curve is the local truncation error the parent note names.

5. The function $f(x,y)$ — a slope at every point

Here is the leap. Normally slope needs a curve first. But a differential equation flips it: it hands you a rule $f(x,y)$ that spits out a slope at any point $(x,y)$ in the plane — before you know the curve.

$y^{\prime }=f(x,y)$

• Plain words: "the steepness of my unknown curve, whenever it passes through the point $(x,y)$, equals $f(x,y)$."
• Picture: at a grid of points draw a tiny arrow with slope $f(x,y)$. This is a direction field — a whole meadow of little arrows.
• Why we need it: $f$ is the only information we get. The curve is hidden; the arrows are given. Euler's job is to trace a path that stays tangent to these arrows.

6. The Initial Value Problem — a starting pin

Arrows fill the whole plane, so infinitely many curves follow them (one through each height). To pick the curve we nail down one point:

$y(x_0)=y_0$

• Plain words: "at input $x_0$, the height is $y_0$" — a pin stuck in the meadow.
• Picture: one dot the curve must pass through; only one arrow-following path threads that dot.
• Why we need it: without the pin there is no unique answer to compute. $x_0,y_0$ are just the coordinates of the pin — the subscript $0$ means "the zeroth / starting one."

7. Subscripts $x_n,y_n$ and the step $h$ — the walker's footprints

Euler walks in equal horizontal strides. Name them:

• $h$ = the step size, the fixed horizontal distance of one stride.

• $x_n$ = the $x$ after $n$ strides: $x_n=x_0+nh$.

• $y_n$ = our estimate of the height there (not the true height — a computed guess).

• Plain words: $n$ is a counter (0, 1, 2, …). $x_n,y_n$ is the $n$-th footprint.

• Picture: a staircase of dots marching right, each $h$ apart horizontally.

• Why we need it: the update formula $y_{n+1}=y_n+hf(x_n,y_n)$ is written entirely in these labels. $y_{n+1}$ = "next height," $y_n$ = "current height."

8. Big-O notation $O(h)$ — measuring smallness

When we say the error is $O(h)$ ("order h") we mean: halve $h$, and the error roughly halves. $O(h^2)$ means: halve $h$, error drops to a quarter.

• Plain words: $O(h^k)$ = "shrinks like the $k$-th power of the step."
• Picture: plot error vs $h$ on a log-log graph — $O(h)$ is a straight line of slope 1, $O(h^2)$ a steeper slope-2 line.
• Why we need it: the entire punchline — "Euler is first order" — is the statement global error is $O(h)$.

9. The leftover symbols: $\xi$, $L$, $M$, $e^{L(x-x_0)}$

These appear only in the error bound. Meet them so they're not scary.

• $\xi$ (Greek "ksi") — some unknown point between $x_n$ and $x_n+h$. Taylor's theorem promises the discarded term is exactly $\frac{h^2}{2}{y}^{\prime \prime }(\xi )$ for one such point; we never need its value, only that it exists. Think "a mystery point living inside the step."
• $y^{\prime \prime }$ — the derivative of the derivative: how fast the slope itself changes, i.e. how much the curve bends (curvature). More bend = tangent leaves the curve faster = bigger error.
• $M$ — a ceiling on the bending: $|y^{\prime \prime }|\le M$ everywhere on the interval. A single number bounding the worst curvature.
• $L$ — the Lipschitz constant (see Lipschitz Continuity): a number saying "if two starting heights differ by $d$, their slopes differ by at most $Ld$." It measures how fast nearby solution curves can spread apart.
• $e^{L(x-x_0)}$ — the exponential growth factor. Errors made early get re-amplified at each later step; the worst-case pile-up grows like this exponential. Bigger $L$ (curves fan out fast) = harsher amplification.

Prerequisite map

This map feeds directly into the parent topic. The Taylor branch is developed in Taylor Series Expansion; the forward-difference view lives in Finite Difference Approximations; the amplification factor is bounded via Lipschitz Continuity; and where small $h$ still fails you meet Numerical Stability, Backward Euler & Implicit Methods, and the higher-order Runge-Kutta Methods.

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