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

$y(t_0) = y_0$

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

$T(t) = T_R + (T_0 - T_R) \; e^{-k t}$

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:

2 s
5 s and
10 s

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

```yml tests: - text: eulersMethod is a function. testString: assert(typeof eulersMethod === 'function', 'eulersMethod is a function.'); - text: eulersMethod(0, 100, 100, 10) should return a number. testString: assert(typeof eulersMethod(0, 100, 100, 10) === 'number', 'eulersMethod(0, 100, 100, 10) should return a number.'); - text: eulersMethod(0, 100, 100, 10) should return 20.0424631833732. testString: assert.equal(eulersMethod(0, 100, 100, 2), 20.0424631833732, 'eulersMethod(0, 100, 100, 10) should return 20.0424631833732.'); - text: eulersMethod(0, 100, 100, 10) should return 20.01449963666907. testString: assert.equal(eulersMethod(0, 100, 100, 5), 20.01449963666907, 'eulersMethod(0, 100, 100, 10) should return 20.01449963666907.'); - text: eulersMethod(0, 100, 100, 10) should return 20.000472392. testString: assert.equal(eulersMethod(0, 100, 100, 10), 20.000472392, 'eulersMethod(0, 100, 100, 10) should return 20.000472392.'); ```