---
title: Euler method
id: 59880443fb36441083c6c20e
challengeType: 5
forumTopicId: 302258
---
## Description
Euler's method numerically approximates solutions of first-order ordinary differential equations (ODEs) with a given initial value. It is an explicit method for solving initial value problems (IVPs), as described in the wikipedia page.
The ODE has to be provided in the following form:
- $\frac{dy(t)}{dt} = f(t,y(t))$
with an initial value
To get a numeric solution, we replace the derivative on the LHS with a finite difference approximation:
- $\frac{dy(t)}{dt} \approx \frac{y(t+h)-y(t)}{h}$
then solve for $y(t+h)$:
- $y(t+h) \approx y(t) + h \, \frac{dy(t)}{dt}$
which is the same as
- $y(t+h) \approx y(t) + h \, f(t,y(t))$
The iterative solution rule is then:
- $y_{n+1} = y_n + h \, f(t_n, y_n)$
where $h$ is the step size, the most relevant parameter for accuracy of the solution. A smaller step size increases accuracy but also the computation cost, so it has always has to be hand-picked according to the problem at hand.
Example: Newton's Cooling Law
Newton's cooling law describes how an object of initial temperature $T(t_0) = T_0$ cools down in an environment of temperature $T_R$:
- $\frac{dT(t)}{dt} = -k \, \Delta T$
or
- $\frac{dT(t)}{dt} = -k \, (T(t) - T_R)$
It says that the cooling rate $\frac{dT(t)}{dt}$ of the object is proportional to the current temperature difference $\Delta T = (T(t) - T_R)$ to the surrounding environment.
The analytical solution, which we will compare to the numerical approximation, is
- $T(t) = T_R + (T_0 - T_R) \; e^{-k t}$
## Instructions
Implement a routine of Euler's method and then to use it to solve the given example of Newton's cooling law with it for three different step sizes of:
and to compare with the analytical solution.
Initial values:
- initial temperature $T_0$ shall be
100 °C
- room temperature $T_R$ shall be
20 °C
- cooling constant $k$ shall be
0.07
- time interval to calculate shall be from
0 s to 100 s
## Tests
```yml
tests:
- text: eulersMethod is a function.
testString: assert(typeof eulersMethod === 'function');
- text: eulersMethod(0, 100, 100, 10) should return a number.
testString: assert(typeof eulersMethod(0, 100, 100, 10) === 'number');
- text: eulersMethod(0, 100, 100, 10) should return 20.0424631833732.
testString: assert.equal(eulersMethod(0, 100, 100, 2), 20.0424631833732);
- text: eulersMethod(0, 100, 100, 10) should return 20.01449963666907.
testString: assert.equal(eulersMethod(0, 100, 100, 5), 20.01449963666907);
- text: eulersMethod(0, 100, 100, 10) should return 20.000472392.
testString: assert.equal(eulersMethod(0, 100, 100, 10), 20.000472392);
```
## Challenge Seed
```js
function eulersMethod(x1, y1, x2, h) {
// Good luck!
}
```
## Solution
```js
function eulersMethod(x1, y1, x2, h) {
let x = x1;
let y = y1;
while ((x < x2 && x1 < x2) || (x > x2 && x1 > x2)) {
y += h * (-0.07 * (y - 20));
x += h;
}
return y;
}
```