# Euler’s Method of Numerical Integration

This program for HP 39gs (HP 40gs) applies Euler’s method to the solution of the differential equation Y’=f(X,Y) on the X-interval [A,B]. The program prompts for input of A, B, an initial value Y = Y(A) and the number N of steps. It displays the resulting approximate value for Y(B). The function f is stored in the Function Aplet "F1(X)". To key in the program, Press [SHIFT] [PROGRAM] and give it a name - Type in EU, for example, and press [ENTER] key.

Type in the following program sequence:

**(B-A)/N**

Press {STO>}**H:**

Press [ENTER] key**Y**

Press {STO>}**Z:1**

Press {STO>}**I:E**

Press {STO>}**X:**

Press [ENTER] key**FOR I=1 TO N STEP 1;**

Press [ENTER] key**Z+F1(X)*H**

Press {STO>}**Z:**

Press [ENTER] key**X+H**

Press {STO>}**X:**

Press [ENTER] key**END:**

Press [ENTER] key**MSGBOX Z:**

Press [ENTER] key

The screen should look like this:

Press [SHIFT] [PROGRAM] to return to the program list.

You can download this Euler Aplet for HP39g+, HP39gs or HP40gs. Go to [SHIFT] [PROGRAM] and {RECV} to upload into your calculator.

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How the Euler program work?

Following are variables used...

A = x0 (First value).

B = Final Estimate for Y.

Y = y0 (Initial value of Y).

N = Steps.

H = Size of each step.

X = Increment (of H) after each step.

F1(X) = Differential Equation stored in Symbolic Function.

Z = Result

User provide A, B, Y and N.

User provide the Differential Equation.

Calculate H.

Loop through the steps to calculate Z.

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To check that you have enter the program correctly.

Enter Y into F1(x), so the differential equation in questionis Y’= Y, with solution Y = e^(X)+1. Run the program with A=0, B=0.5,Y=1, N=5.

Press [SYMB]

Enter the following into F1(X):

Press the [Home]

Enter the following variables:

And Press [ENTER].

Where A=0, B=0.5,Y=1, N=5.

Type RUN EU and press [ENTER].

The result will be prompted.

The result should be 2.1168.

Updated On: 15.02.19