--- id: 59880443fb36441083c6c20e title: Euler method challengeType: 5 forumTopicId: 302258 dashedName: euler-method --- # --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 this article. The ODE has to be provided in the following form:

$\frac{dy(t)}{dt} = f(t,y(t))$

with an initial value

$y(t_0) = y_0$

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 $T_R$:

$\frac{dT(t)}{dt} = -k \, \Delta T$

$\frac{dT(t)}{dt} = -k \, (T(t) - T_R)$

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 use it to solve the given example of Newton's cooling law for three different step sizes of:

2 s
5 s and
10 s

and 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

First parameter to the function is initial time, second parameter is initial temperature, third parameter is elapsed time and fourth parameter is step size. # --hints-- `eulersMethod` should be a function. ```js assert(typeof eulersMethod === 'function'); ``` `eulersMethod(0, 100, 100, 2)` should return a number. ```js assert(typeof eulersMethod(0, 100, 100, 2) === 'number'); ``` `eulersMethod(0, 100, 100, 2)` should return 20.0424631833732. ```js assert.equal(eulersMethod(0, 100, 100, 2), 20.0424631833732); ``` `eulersMethod(0, 100, 100, 5)` should return 20.01449963666907. ```js assert.equal(eulersMethod(0, 100, 100, 5), 20.01449963666907); ``` `eulersMethod(0, 100, 100, 10)` should return 20.000472392. ```js assert.equal(eulersMethod(0, 100, 100, 10), 20.000472392); ``` # --seed-- ## --seed-contents-- ```js function eulersMethod(x1, y1, x2, h) { } ``` # --solutions-- ```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; } ```