IB Mathematics Applications & Interpretation Topic…

Why Euler's Method Exists: A Numerical Route When No Formula Will Do

Many differential equations that come out of real models cannot be solved with a tidy formula.
Euler's method sidesteps that by stepping forward in small straight-line hops, using the gradient the equation gives at each point.
It is the slope field turned into arithmetic: instead of drawing segments, you walk along them a little at a time.
The trade-off is that the answer is an approximation, and its accuracy depends on how small your steps are.

Analogy

It is like navigating in fog by compass: you can only see the direction right where you stand.
So you take a short step in that direction, check the new direction, and repeat.

Euler's Method: Walking Along The Tangent

Start from a known point $(x_0, y_0)$ and a chosen step size $h$.
Because the gradient there is $f(x_0, y_0)$, the tangent-line estimate of the next $y$ is the old $y$ plus slope times step. $$ x_{n+1} = x_n + h, \qquad y_{n+1} = y_n + h\,f(x_n, y_n) $$

Example

For $\dfrac{dy}{dx} = x + y$ with $y(0) = 1$ and $h = 0.1$:

First step (shown in full):
- Slope $= f(0,1) = 1$,
- so $y_1 = 1 + 0.1\times 1 = 1.1$ at $x_1 = 0.1$.
Second step:
- Slope $= f(0.1, 1.1) = 1.2$,
- so $y_2 = 1.1 + 0.1\times 1.2 = 1.22$ at $x_2 = 0.2$.

n	$x_n$	$y_n$	slope f	$y_(n+1)$
0	0	1	1	1.1
1	0.1	1.1	1.2	1.22
2	0.2	1.22	-	-

Note

The recurrence for Euler's method is given in the formula booklet.
Always show the first iteration in full, because that is where the method mark is earned.

Example question

Given $\dfrac{dy}{dx} = 2x - y$ with $y = 3$ at $x = 0$, use Euler's method with step size $h = 0.5$ to estimate $y$ when $x = 1$.

Solution

Step 1:
- Slope $= 2(0) - 3 = -3$,
- so $y_1 = 3 + 0.5(-3) = 1.5$ at $x_1 = 0.5$.
Step 2:
- Slope $= 2(0.5) - 1.5 = -0.5$,
- so $y_2 = 1.5 + 0.5(-0.5) = 1.25$ at $x_2 = 1$.
Estimate: $$y(1) \approx 1.25$$

Accuracy And Step Size

Each step replaces a curve by its tangent, so a small error creeps in and then accumulates as the steps go on.
A smaller step size $h$ gives a more accurate estimate, at the cost of more steps to reach the same $x$.
The errors tend to build in one direction, so Euler's estimate usually drifts steadily above or below the true solution.

Common Mistake

Euler's answer is an approximation, so do not quote it as exact.
When the true solution curves upward (concave up), the tangent steps sit below it, so Euler underestimates.
Halving $h$ typically halves the error.

Coupled Systems: Two Quantities Changing Together

Sometimes two quantities affect each other's rates, such as the numbers of predators and prey in an ecosystem.
This is a coupled system, two differential equations sharing the variable $t$: $$\dfrac{dx}{dt} = f_1(x,y)$ and $\dfrac{dy}{dt} = f_2(x,y)$$
Euler handles it by running both recurrences in parallel, using the current $x$ and $y$ to update each. $$ x_{n+1} = x_n + h\,f_1(x_n, y_n), \qquad y_{n+1} = y_n + h\,f_2(x_n, y_n) $$

Example

$\dfrac{dx}{dt} = y$, $\dfrac{dy}{dt} = -x$, starting at $(1, 0)$ with $h = 0.1$.
One step gives $x_1 = 1 + 0.1(0) = 1$ and $y_1 = 0 + 0.1(-1) = -0.1$ at $t = 0.1$.

Exam technique

Use the same booklet recurrence, once for each equation, updating both from the current values before advancing $t$.
Show the first iteration of each equation in full, and keep a neat table of $t$, $x$ and $y$.
This same parallel-recurrence idea powers second-order equations in 5.18, once they are rewritten as a coupled pair.

Example question

A system satisfies $\dfrac{dx}{dt} = 3x - y$ and $\dfrac{dy}{dt} = x + y$, with $x = 1$ and $y = 2$ at $t = 0$. Use Euler's method with $h = 0.1$ to estimate $x$ and $y$ at $t = 0.1$.

Solution

$\dfrac{dx}{dt} = 3(1) - 2 = 1$, so $x_1 = 1 + 0.1(1) = 1.1$.
$\dfrac{dy}{dt} = 1 + 2 = 3$, so $y_1 = 2 + 0.1(3) = 2.3$.
At $t = 0.1$: $$x \approx 1.1$, $y \approx 2.3$$

Single Versus Coupled At A Glance

Feature	Single equation	Coupled system
Unknowns	One (y)	Two (x and y)
Recurrences per step	One	Two, run together
Updates use	Current x, y	Current x and y for both

Active recall

Write the Euler recurrence for $y_{n+1}$ and say where it comes from.
Why does a smaller step size $h$ improve accuracy, and what is the cost?
If the true solution is concave up, does Euler over- or under-estimate, and why?
How is a coupled system stepped forward with Euler's method?
Which later topic reuses the coupled-recurrence idea, and how?

Why Euler's Method Exists: A Numerical Route When No Formula Will Do

Many differential equations that come out of real models cannot be solved with a tidy formula.
Euler's method sidesteps that by stepping forward in small straight-line hops, using the gradient the equation gives at each point.
It is the slope field turned into arithmetic: instead of drawing segments, you walk along them a little at a time.
The trade-off is that the answer is an approximation, and its accuracy depends on how small your steps are.

Analogy

It is like navigating in fog by compass: you can only see the direction right where you stand.
So you take a short step in that direction, check the new direction, and repeat.

Euler's Method: Walking Along The Tangent

Start from a known point $(x_0, y_0)$ and a chosen step size $h$.
Because the gradient there is $f(x_0, y_0)$, the tangent-line estimate of the next $y$ is the old $y$ plus slope times step. $$ x_{n+1} = x_n + h, \qquad y_{n+1} = y_n + h\,f(x_n, y_n) $$

Example

For $\dfrac{dy}{dx} = x + y$ with $y(0) = 1$ and $h = 0.1$:

First step (shown in full):
- Slope $= f(0,1) = 1$,
- so $y_1 = 1 + 0.1\times 1 = 1.1$ at $x_1 = 0.1$.
Second step:
- Slope $= f(0.1, 1.1) = 1.2$,
- so $y_2 = 1.1 + 0.1\times 1.2 = 1.22$ at $x_2 = 0.2$.

n	$x_n$	$y_n$	slope f	$y_(n+1)$
0	0	1	1	1.1
1	0.1	1.1	1.2	1.22
2	0.2	1.22	-	-

Note

The recurrence for Euler's method is given in the formula booklet.
Always show the first iteration in full, because that is where the method mark is earned.

Example question

Given $\dfrac{dy}{dx} = 2x - y$ with $y = 3$ at $x = 0$, use Euler's method with step size $h = 0.5$ to estimate $y$ when $x = 1$.

Solution

Step 1:
- Slope $= 2(0) - 3 = -3$,
- so $y_1 = 3 + 0.5(-3) = 1.5$ at $x_1 = 0.5$.
Step 2:
- Slope $= 2(0.5) - 1.5 = -0.5$,
- so $y_2 = 1.5 + 0.5(-0.5) = 1.25$ at $x_2 = 1$.
Estimate: $$y(1) \approx 1.25$$

Accuracy And Step Size

Each step replaces a curve by its tangent, so a small error creeps in and then accumulates as the steps go on.
A smaller step size $h$ gives a more accurate estimate, at the cost of more steps to reach the same $x$.
The errors tend to build in one direction, so Euler's estimate usually drifts steadily above or below the true solution.

Common Mistake

Euler's answer is an approximation, so do not quote it as exact.
When the true solution curves upward (concave up), the tangent steps sit below it, so Euler underestimates.
Halving $h$ typically halves the error.

Coupled Systems: Two Quantities Changing Together

Sometimes two quantities affect each other's rates, such as the numbers of predators and prey in an ecosystem.
This is a coupled system, two differential equations sharing the variable $t$: $$\dfrac{dx}{dt} = f_1(x,y)$ and $\dfrac{dy}{dt} = f_2(x,y)$$
Euler handles it by running both recurrences in parallel, using the current $x$ and $y$ to update each. $$ x_{n+1} = x_n + h\,f_1(x_n, y_n), \qquad y_{n+1} = y_n + h\,f_2(x_n, y_n) $$

Example

$\dfrac{dx}{dt} = y$, $\dfrac{dy}{dt} = -x$, starting at $(1, 0)$ with $h = 0.1$.
One step gives $x_1 = 1 + 0.1(0) = 1$ and $y_1 = 0 + 0.1(-1) = -0.1$ at $t = 0.1$.

Exam technique

Use the same booklet recurrence, once for each equation, updating both from the current values before advancing $t$.
Show the first iteration of each equation in full, and keep a neat table of $t$, $x$ and $y$.
This same parallel-recurrence idea powers second-order equations in 5.18, once they are rewritten as a coupled pair.

Example question

A system satisfies $\dfrac{dx}{dt} = 3x - y$ and $\dfrac{dy}{dt} = x + y$, with $x = 1$ and $y = 2$ at $t = 0$. Use Euler's method with $h = 0.1$ to estimate $x$ and $y$ at $t = 0.1$.

Solution

$\dfrac{dx}{dt} = 3(1) - 2 = 1$, so $x_1 = 1 + 0.1(1) = 1.1$.
$\dfrac{dy}{dt} = 1 + 2 = 3$, so $y_1 = 2 + 0.1(3) = 2.3$.
At $t = 0.1$: $$x \approx 1.1$, $y \approx 2.3$$

Single Versus Coupled At A Glance

Feature	Single equation	Coupled system
Unknowns	One (y)	Two (x and y)
Recurrences per step	One	Two, run together
Updates use	Current x, y	Current x and y for both

Active recall

Write the Euler recurrence for $y_{n+1}$ and say where it comes from.
Why does a smaller step size $h$ improve accuracy, and what is the cost?
If the true solution is concave up, does Euler over- or under-estimate, and why?
How is a coupled system stepped forward with Euler's method?
Which later topic reuses the coupled-recurrence idea, and how?

IB Mathematics Applications & Interpretation AHL 5.16—eulers Method for 1st Order Des Notes

IB Mathematics Applications & Interpretation AHL 5.16—eulers Method for 1st Order Des Notes

Why Euler's Method Exists: A Numerical Route When No Formula Will Do

Euler's Method: Walking Along The Tangent

Accuracy And Step Size

Coupled Systems: Two Quantities Changing Together

Single Versus Coupled At A Glance

Why Euler's Method Exists: A Numerical Route When No Formula Will Do

Euler's Method: Walking Along The Tangent

Accuracy And Step Size

Coupled Systems: Two Quantities Changing Together

Single Versus Coupled At A Glance

IB Mathematics Applications & Interpretation AHL 5.16—eulers Method for 1st Order Des Notes

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

IB Mathematics Applications & Interpretation AHL 5.16—eulers Method for 1st Order Des Notes

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

Why Euler's Method Exists: A Numerical Route When No Formula Will Do

Euler's Method: Walking Along The Tangent

Accuracy And Step Size

Coupled Systems: Two Quantities Changing Together

Single Versus Coupled At A Glance

Why Euler's Method Exists: A Numerical Route When No Formula Will Do

Euler's Method: Walking Along The Tangent

Accuracy And Step Size

Coupled Systems: Two Quantities Changing Together

Single Versus Coupled At A Glance