Euler's numerical method (HL)

Studied in greater depth in AI HL than AA HL, Euler’s method is made easy here on TI 84, TI Nspire, Casio fx-9860GIII, Casio fx-CG50, HP Prime, and NumWorks.

For most calculator models, both the main method (“recursive”) and an alternative method (“chaining”) are discussed.

Also at the end is a second order DE for Math AI HL, along with plotting the coordinates found.

Re-written on 2026-03-17. Updated 2026-07-22 with additional Nspire spreadsheet optimization

Theory
GDC Procedure
- chaining ( $y$ then $x$ )
- recursive ( $x$ then $y$ )
Second order and coupled DE (AI HL only)

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

\frac{\d y}{\d x} = f(x, y)

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

x_{n+1} = x_n + h

y_{n+1} = y_n + hf(x_n, y_n)

Note that $y_{n+1}$ depends on both $y_n$ and $x_n$ , but $x_{n+1}$ depends only on $x_n$ .

$f(x_n, y_n)$ is the estimate for the gradient of the line. $h$ is $\Delta x$ . So the equation for $y$ is saying

\Delta y_{n+1} = \Delta x_n \cdot \text{gradient}_n

$h$ is positive or negative depending on if moving to the right or left.

Note: IB may ask for $x$ or $y$ . You may also get

\frac{\d x}{\d t} = f(t, x)

and other variables. If that happens treat the $x$ as $y$ and the $t$ as $x$ .

GDC Procedure

Example: Given that

\frac{\d y}{\d x} = \frac{y^2}{1+x},\;\; y(0) = 1

use Euler’s method with a step size of $h= 0.1$ , estimate $y$ when $x = 0.5$

Solution:

\begin{align*} x_{n+1} &= x_n + 0.1 &\text{GDC uses } a_{n+1} \\ y_{n+1} &= y_n + 0.1\left(\frac{y_n^2}{1+x_n}\right) &\text{GDC uses } b_{n+1} \end{align*}

Commentary: Using question variables, write down the Euler’s method formula with the derivative substituted in. To avoid data entry errors, write down the correspondence between calculator variables (usually u_n, v_n or a_n, b_n). Also good idea to wrap the derivative in parentheses ( ).

$x$	$y$
0	1
0.1	1.1
…	…
0.5	1.623

$y \approx 1.62$ when $x = 0.5 \qed$

Tip: In the past, IB has asked for over 15 iterations of Euler’s method. I recommend to write down Euler’s method formula with substitutions, and a correct coordinate for each remaining mark, and always include the starting point, first and last iterations.

Warning: Check if question wants a point or just the $y$ -value. Answer the question as asked.

Here’s how to do it on each graphing calculator. To plot the solution, check out Second order and coupled DE (AI HL only).

Generally, we can either use calculation chaining or recursive sequences. Try both methods across multiple questions, and pick what you prefer.

chaining ( $y$ then $x$ )

This method is by Dr. William J. Larson for TI 84 Plus, then I expanded it for several other graphic display calculator (GDC) models as well.

Because the new Y depends on the old value of X, we update Y before X. We should also see the new values. We need to chain everything in a single calculation, so Enter or EXE can re-run the last calculation and update both variables.

Initialize X, Y, H. Storing H (step size) is optional.
0→X 1→Y 0.1→H
TI uses : Casio uses ⤶ HP uses ; to chain calculations.

separate:
Y+H(Y^2/(1+X))→Y X+H→X {X,Y}

chained: Y+H(Y^2/(1+X))→Y:X+H→X:{X,Y}
Press Enter or EXE button until we see the desired X. Note: No need to re-enter the expression.

Warning: If you mess up on step 2, you must redo step 1.

Example: Given that

\frac{\d y}{\d x} = \frac{y^2}{1+x},\;\; y(0) = 1

use Euler’s method with a step size of $h= 0.1$ , estimate $y$ when $x = 0.5$

TI 84 Plus CE TI Nspire CX Casio fx-9860GIII HP Prime

Store sto→ the initial values in X and Y alpha 1. Optionally we can also store the step size in H.

Chain the commands using : alpha • to update Y, X, and show the new coordinates using {, 2nd (, } 2nd ) and comma ❟.

entering the expression

Press enter until we see {0.5 ... for $x = 0.5$ .

a few more points

our solution

purpose	symbol	button(s)
store	`→`	sto→
calculation separator	`:`	alpha •
curly braces	`{`, `}`	2nd (, 2nd )
comma	`,`	❟
nice fractions		alpha y= `1:n/d`
get next point		enter

The built in euler( is disabled during IB exams.

Store ctrl var the initial values in x and y. Optionally we can also store the step size in h.
Chain the commands using : from ?!▸ to update y, x, and show the new coordinates using {, } ctrl )
Press enter until we see {0.5 ... for $x = 0.5$ .

Entering above expressions on TI Nspire CX

Results for Euler's method on TI Nspire CX

purpose	symbol	button(s)
store	`→`	ctrl var
calculation separator	`:`	from ?!▸
curly braces	`{`, `}`	ctrl )
comma	`,`	❟
nice fractions		from □{□
get next point		enter

For this particular method, we need Line or Linear mode. This means more brackets when entering math. This shortcoming makes chaining on Casio not ideal.

Images are from Casio fx-9860GIII. Very similar results on fx-9750GIII and fx-CG50.

In Run/Mat, set up Shift MENU, change to Line (Linear) mode F2 at the top.

Change to Linear I/O temporarily
Store → the initial values in X and Y. Optionally we can also store the step size in H.

initialize X, Y, H on Casio fx-9860GIII
Chain the commands using newline Shift EXE to update Y, X, and show the new coordinates using curly braces Shift ×, Shift ÷ and comma ❟.

Euler's method main step on Casio fx-9860GIII
Each result is shown in its own screen as a List. Copy it down.

coordinates after 1 iteration
Press EXE twice for the next point. Repeat. Continue until we see 0.5 on top for $x = 0.5$ .

final point in Euler's method

This is how the main screen should look.

main screen at the end
In set up Shift MENU, change back to Math I/O. when done with Euler’s method.

purpose	symbol	button(s)
store	`→`	→
calculation separator	`⤶`	Shift EXE
curly braces	`{`, `}`	Shift ×, Shift ÷
comma	`,`	❟
get next point		EXE

In ⌂ (home), store Shift EEX , the initial values in X and Y. Optionally we can also store the step size in H.
Chain the commands using ; from ALPHA + to update Y, X, and show the new coordinates in one step using curly braces Shift 8 and comma
Press Enter until we until we see {0.5 ... for $x = 0.5$ .

Euler's method using calculation chaining on HP Prime

purpose	symbol	button(s)
store	`▶`	Shift EEX
calculation separator	`;`	ALPHA +
curly braces	`{`, `}`	Shift 8
comma	`,`	❟
get next point		Enter

recursive ( $x$ then $y$ )

This method defines two sequences, where u(n+1) depends on u(n) and v(n+1) depends on both u(n) (ie $x$ ) and v(n) (ie $y$ ). We would create u then v ( $x$ then $y$ ).

Activate recursive sequence mode/app. Use u(n+1) to define recursive sequences starting from n=1.
Define
u(1)=0 u(n+1)=u(n)+0.1 v(1)=1 v(n+1)=v(n)+0.1v(n)^2/(1+u(n))
Check table.

When the sequences are defined recursively, it does not matter whether we start at n = 0 or n = 1, and just use GDC default.

Example: Given that

\frac{\d y}{\d x} = \frac{y^2}{1+x},\;\; y(0) = 1

use Euler’s method with a step size of $h= 0.1$ , estimate $y$ when $x = 0.5$

TI 84 Plus CE TI Nspire CX Casio fx-9860GIII HP Prime NumWorks

On non-CE calculators, follow the instructions below but write (n-1) each time instead of (n).

In mode instead of FUNC or FUNCTION choose SEQ.

use SEQ mode
In y=, highlight SEQ(n+1). Set nMin to 1 (default). Define u(n+1), u(1), v(n+1) and v(1).

define the two sequences; both are recursive.
In 2nd graph, view the table. Settings can be changed in 2nd window.

table of values
To view the graph, see #Euler: coupled DE and use uv in window
Change back to FUNC or FUNCTION mode when done.

purpose	symbol	button(s)
sequence mode	`SEQ`	mode
recursive sequence	`TYPE:seq(n+1)`	use arrow keys
set `n`	`nMin=1`	use arrow keys
sequences	`u`, `v`	2nd 7, 2nd 8
nice fractions		alpha y= `1:n/d`
set table		2nd window
view table		graph

In Lists & Spreadsheet app (via ctrl doc), enter 0 in A1 and 1 in B1. In A2, enter a1 + 0.1. In B2, enter
$=b1 + 0.1\left(\frac{b1^2}{1+a1}\right)$

initialize the two columns and set formula in A2

initialize B2 applying formula for Euler's method
Highlight cell A2. Press shift then right arrow to highlight both A2 and B2.

use shift to select multiple cells
Go to menu 3:Data 3:Fill or ctrl menuFill. Use down arrow to choose which rows to paste the formula. enter to confirm.

highlight A2 and B2 while dragging

select cells to paste formula

enter to confirm

Warning: When you see ..., the GDC truncated the value (rounding it down) instead of rounding normally. Highlight the value to confirm.

purpose	button(s)
spreadsheet	ctrl doc `Lists & Spreadsheet` app
drag formula	menu `3:Data` `3:Fill`, or ctrl menu`Fill`
cell name	`a1` for cell `A1`

Images are from Casio fx-9860GIII. Very similar results on fx-9750GIII and fx-CG50.

In MENU RECUR or Recursion app, in F3 (TYPE) choose F2:an+1=Aan+Bn+C
Define the two sequences recursively using the F4 (n,an) tab.

define the two sequences in an+1 and bn+1
In F5 (SET), change END to show enough terms, and set a0 and b0 to the initial values.

main screen at the end
View table in F6 (TABL). Scroll for the desired value.

table of values

purpose	symbol	button(s)
recursion app		MENU, under `RECUR`
recursive sequence	`F2:an+1=Aan+Bbn+C`	F3 (`TYPE`)
set `n`	`Start`, `End`	F5 (`SET`)
sequences	`an`. `bn`	F4 (`Name`)
set table		F5 (`SET`)
view table		F6 (`TABL`)

In Apps , use Sequence app near the end.
Choose U(N+k) for recursive sequences. Define U1(n+1) and U2(n+1). Use on-screen buttons as needed.

Define U1(n+1) and U2(n+1) in Sequences app
In Num ⊞ to view the table. Settings can be changed in Shift Num ⊞ . Go back to Symb ✗ to change the sequences.

table of values

purpose	symbol	button(s)
sequence mode		Apps , `Sequence` app
recursive sequence	`U(N+k)`	Symb ✗ , on screen
set `n`	`Start N`	Symb ✗ , on screen
sequences	`U`	touch screen, or ALPHA 4
set table		Shift Num ⊞
view table		Num ⊞

In ⌂ (home), use Sequences app.
Add a sequence: u_n Explicit expression. Set u_n = 0 + 0.1(n-1).
Add a sequence: u_n+1 Recursive first order. Set v_n+1 and v1. The variables are from var, Sequences.

define in Sequences app
Use arrow keys to move to Table tab to view the table and change settings.

viewing the table

Note you can also do recursive for both, it’s just that for purpose of taking screenshots I did explicit and recursive.

purpose	symbol	button(s)
sequences mode		⌂ (home), `Sequences` app
recursive sequence	`u_n+1 Recursive first order`	`Add a sequence`
set `n`	`First term index`	highlight three dots after sequence is defined
sequences	`u`, `v`	var, `sequences`
set/view table		`Table` tab

Second order and coupled DE (AI HL only)

Pretty much the same thing, but we have two derivatives and require two to three variables. Only the recursive method will be explored, as the chaining method needs three to four variables.

Example: M25 P2 TZ2 HL 6 (d), modified

A financial analyst models the change in value of one share, x dollars at time t minutes, after a report is released. She uses the differential equation

\frac{\d^2 x}{\d t^2} + 4\frac{\d x}{\d t} + 3x = x \sin t

This equation can be written as the coupled differential equations

\frac{\d x}{\d t} = y \\ \frac{\d y}{\d t} = -3x - 4y + x \sin t

Initially $x = 0$ and $\displaystyle \frac{\d x}{\d t} = -1$ .

(d) Use Euler’s method with a $t$ -interval of $0.1$ to predict the value of $x$ when $t = 1$ .

“Initially” means $t = 0$ . This questions happens to use the same $h = 0.1$ as above, but that is just a coincidence.

\begin{align*} t_{n+1} &= t_n + 0.1 &\text{GDC uses } a_{n+1} \\ x_{n+1} &= x_n + 0.1\left(y_n\right) &\text{GDC uses } b_{n+1} \\ y_{n+1} &= y_n + 0.1\left(-3x_n - 4y_n + x_n \sin t_n\right) &\text{GDC uses } c_{n+1} \end{align*}

Commentary: Question does not usually provide the coupled DE for you. Using question variables, write down the Euler’s method formula with the derivatives substituted in. To avoid data entry errors, write down the correspondence between calculator variables (usually u_n, v_n or a_n, b_n). Also good idea to wrap the derivatives in parentheses ( ).

Tip: Using recursive form above means you never have to worry about your minimum n, and can always use the GDC default, whether that is 0 or 1.

The values

$t$	$x$	$y$
0	0	-1
0.1	-0.1	-0.6
…	…	…
1.0	-0.1718	0.1043

When $t = 1$ , $x$ is approx $-0.172 \qed$

Tip: ensure your GDC is in radians when seeing trig functions but not the $\degree$ (degree) symbol.

These instructions assume you are familiar with ones from recursive x then y above.

TI 84 Plus CE TI Nspire CX Casio fx-CG50 HP Prime NumWorks

In mode instead of FUNC or FUNCTION choose SEQ. In window, use SEQ(n+1), and type up the above recursive relations.
Check 2nd graph

Table of values

To graph the solution

In window set appropriate n, X and Y values.

Table of values
In 2nd window, change top line to vw. For a single (non-coupled DE), choose uv.

Select the two sequences to plot
Graph graph.

DE solution y plotted against x

In Lists & Spreadsheet app (via ctrl doc), enter the boundary conditions in the first row. Enter the formulas into A1, B2 and C2. We also need to name the columns to view a plot in step 4.
Highlight cell A2. Press shift, right arrow, shift, right arrow, to highlight A2, B2, and C2.

press shift after each additional cell
Go to menu 3:Data 3:Fill or ctrl menuFill. Use down arrow to choose which rows to paste the formula. enter to confirm.

choose menu > data > fill

select cells to fill

enter to confirm; coordinates in B11 and C11

Warning: When you see ..., the GDC truncated the value (rounding it down) instead of rounding normally. Highlight the value to confirm.

To graph the solution in $xy$ -plane, add a Data & Statistics tab via ctrl doc. Select the x and y columns named earlier.

DE solution y plotted against x

In MENU RECUR or Recursion app, in F3 (TYPE) choose F2:an+1=Aan+Bn+C. Define the above sequences using the F4 (n,an) tab.
In F5 (SET), change END to show enough terms, and set a0, b0, c0 to the initial values.
View table in F6 (TABLE). Scroll for the desired value.

Warning: GDC cuts the number at $-0.171$ and we need to highlight value to see it is $-0.172$

To graph the solution

In Shift F3, change the X and Y zooms.
Exit. In F3 (PHASE), select b-c for the second and third lists. For single DE, select a-b instead.
Graph using F6.

In Apps , use Sequence app near the end.
Choose U(N+k) for recursive sequences. Define U1(n+1) and U2(n+1). Use on-screen buttons as needed.
In Num ⊞ to view the table. Settings can be changed in Shift Num ⊞ . Go back to Symb ✗ to change the sequences.

To graph the solution

In Apps , select Statistics 2Var. Make C1 into U2(X), and C2 into U3(X), with X maximum set to upper limit, which here is 11.
In Symb ✗ , ensure S1 has C1 and C2 beside it.
Plot using Plot . In Menu, Zoom, choose B AUTOSCALE.

In ⌂ (home), use Sequences app. Add three sequences of type Recursive first order. This allows u, v, w to be stored in var, Sequences.
Use arrow keys to move to Table tab to view the table and change settings.

To graph the solution

In ⌂ (home), use Regression app. Highlight X1 then press EXE. Select Fill with a formula. Modify the {k}_k≤10 template to add {v_k} to X1 and Y1. The variable k can be entered using x,n,t. We want to go up to n=11. Do the same to add {w_k} to Y1.

$X1 = {v_k}_k≤11$

$Y1 = {w_k}_k≤11$
Use arrow keys and select Graph. By default, the zooms should be set for you.

Previous HL: Differential equations overview (HL)

Next HL: Separable differential equations (HL)

Euler's numerical method (HL)

Contents

Theory

GDC Procedure

chaining (yyy then xxx)

recursive (xxx then yyy)

Second order and coupled DE (AI HL only)

chaining ( $y$ then $x$ )

recursive ( $x$ then $y$ )