EULER'S METHOD: WHAT IT IS FOR, PROCEDURE AND EXERCISES - MATHS - 2025

The Euler method is the most basic and simple procedures used to find numerical solutions approximate to an ordinary differential equation of the first order, provided that the initial condition is known.

An ordinary differential equation (ODE) is the equation that relates an unknown function of a single independent variable with its derivatives.

Successive approximations by Euler's method. Source: Oleg Alexandrov

If the largest derivative that appears in the equation is of degree one, then it is an ordinary differential equation of the first degree.

The most general way to write an equation of the first degree is:

x = x ₀

y = y ₀

What is Euler's method?

The idea of ​​Euler's method is to find a numerical solution to the differential equation in the interval between X ₀ and X _f.

First, the interval is discretized in n + 1 points:

x ₀, x ₁, x ₂, x ₃ …, x _n

Which are obtained like this:

x _i = x ₀ + ih

Where h is the width or step of the subintervals:

With the initial condition, then it is also possible to know the derivative at the beginning:

y '(x _o) = f (x _o, y _o)

This derivative represents the slope of the tangent line to the curve of the function y (x) precisely at the point:

Ao = (x _o, y _o)

Then an approximate prediction of the value of the function y (x) is made at the following point:

y (x ₁) ≈ y ₁

y _{1 =} y _o + (x ₁ - x _o) f (x _o, y _o) = y _{o +} hf (x _o, y _o)

The next approximate point of the solution has then been obtained, which would correspond to:

A ₁ = (x ₁, y ₁)

The procedure is repeated to obtain the successive points

A ₂, A ₃ …, x _n

In the figure shown at the beginning, the blue curve represents the exact solution of the differential equation, and the red one represents the successive approximate points obtained by the Euler procedure.

Solved exercises

Exercise 1

I) Let the differential equation be:

With the initial condition x = a = 0; and _a = 1

Using Euler's method, get an approximate solution of y at the coordinate X = b = 0.5, subdividing the interval into n = 5 parts.

Solution

The numerical results are summarized as follows:

From which it is concluded that the solution Y for the value 0.5 is 1.4851.

Note: Smath Studio, a free program for free use, has been used to perform the calculations.

Exercise 2

II) Continuing with the differential equation from exercise I), find the exact solution and compare it with the result obtained by Euler's method. Find the error or difference between the exact and the approximate result.

Solution

The exact solution is not very difficult to find. The derivative of the function sin (x) is known to be the function cos (x). Therefore the solution y (x) will be:

y (x) = sin x + C

For the initial condition to be fulfilled and (0) = 1, the constant C must be equal to 1. The exact result is then compared with the approximate one:

It is concluded that in the calculated interval, the approximation has three significant figures of precision.

Exercise 3

III) Consider the differential equation and its initial conditions given below:

y '(x) = - y ²

With the initial condition x ₀ = 0; and ₀ = 1

Use Euler's method to find approximate values ​​of the solution y (x) on the interval x =. Use step h = 0.1.

Solution

Euler's method is very suitable for use with a spreadsheet. In this case we will use the geogebra spreadsheet, a free and open-source program.

The spreadsheet in the figure shows three columns (A, B, C) the first is the variable x, the second column represents the variable y, and the third column is the derivative y '.

Row 2 contains the initial values ​​of X, Y, Y '.

The value step 0.1 has been placed in the absolute position cell ($ D $ 4).

The initial value of y0 is in cell B2, and y1 is in cell B3. To calculate y ₁ the formula is used:

y _{1 =} y _o + (x ₁ - x _o) f (x _o, y _o) = y _{o +} hf (x _o, y _o)

This spreadsheet formula would be Number B3: = B2 + $ D $ 4 * C3.

Similarly y2 would be in cell B4 and its formula is shown in the following figure:

The figure also shows the graph of the exact solution, and the points A, B,…, P of the approximate solution by Euler's method.

Newtonian dynamics and Euler's method

Classical dynamics was developed by Isaac Newton (1643 - 1727). The original motivation of Leonard Euler (1707 - 1783) to develop his method, was precisely to solve the equation of Newton's second law in various physical situations.

Newton's second law is usually expressed as a differential equation of the second degree:

Where x represents the position of an object at time t. Said object has a mass m and is subjected to a force F. The function f is related to force and mass as follows:

To apply Euler's method the initial values ​​of time t, velocity v and position x are required.

The following table explains how starting from initial values ​​t1, v1, x1, an approximation of the velocity v2 and the position x2 can be obtained, at the instant t2 = t1 + Δt, where Δt represents a small increase and corresponds to the step in the method of Euler.

Exercise 4

IV) One of the fundamental problems in mechanics is that of a block of mass M tied to a spring (or spring) of elastic constant K.

Newton's second law for this problem would look like this:

In this example, for simplicity we will take M = 1 and K = 1. Find approximate solutions to the position x and the velocity v by Euler's method on the time interval by subdividing the interval into 12 parts.

Take 0 as the initial instant, initial velocity 0, and initial position 1.

Solution

The numerical results are shown in the following table:

The graphs of the position and velocity between times 0 and 1.44 are also displayed.

Proposed exercises for home

Exercise 1

Use a spreadsheet to determine an approximate solution using Euler's method for the differential equation:

y '= - Exp (-y) with the initial conditions x = 0, y = -1 in the interval x =

Start with a 0.1 step. Plot the result.

Exercise 2

Using a spreadsheet, find numerical solutions to the following quadratic equation, where y is a function of the independent variable t.

y '' = - 1 / y² with the initial condition t = 0; and (0) = 0.5; y '(0) = 0

Find the solution in the interval using a step of 0.05.

Plot the result: y vs t; y 'vs t

References

1. Eurler method Taken from wikipedia.org
2. Euler solver. Taken from en.smath.com