Euler's numerical method (HL)

One of the few calculus topics that is studied in greater depth in AI HL than AA HL.

Contents

• Theory
• Practice

Theory

In interpretations of derivatives, it was mentioned that a smooth curve can be approximated by a series of jagged lines segments as each line segment becomes small.

This is exactly what we do in Euler’s method.

Given some differential equation

and some initial point $(x_0, y_0)$, and a step size $h$, then each successive value of $x$, and $y$ is

Note that each iteration depends on only the previous iteration. $f(x_n, y_n)$ is the estimate for the gradient of the line. $h$ is $\Delta x$. So the equation for $y$ is saying

On some calculators, you instead use

In both cases, start with $x_0, y_0$, and after $k$ iterations, return $y_k$ as a final answer. Questions could also sometimes ask for $(x_k, y_k)$ instead.

Practice

Use sequence mode or recursion mode or spreadsheet on your calculator to implement a solution. As your calculator gets reset before your exams, it is good practice to make the math wrap as opposed to display in one line.

Given that

Use a step size of $h= 0.1$, estimate $y$ when $x = 0.5$

On TI-84 Plus, u, v, w are available via 2ND + 7, 2ND + 8, 2ND + 9. Here we presume the lists or sequences are named u, v, w etc. It depends on specific model of your calculator.

1. Set calculator to sequences mode.
2. In y=, set nMin to 0.
3. Define u(nMin) = 0, u(n) = 0.1*n, for a list of $x_n$
4. Define v(nMin) = 1, v(n) = v(n-1) + 0.1*(v(n-1)^2/(1+u(n-1))), for a list of $y_n$
5. See table of values where n is 5, and u(n) is .5, and find v(n) to be 1.6225 $\qed$

Typically, writing down a full, correct list of each $y_n$ is sufficient for full marks.

In the above example, an extra set of parentheses were used after h*, which is a good practice in case $f(x, y)$ is not a fraction, though not necessary in this particular question.