chore(i18n,learn): processed translations (#44851)

curriculum/challenges/japanese/10-coding-interview-prep/rosetta-code/euler-method.md

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+---
+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](https://www.freecodecamp.org/news/eulers-method-explained-with-examples/ "news: Euler's Method Explained with Examples").
+
+The ODE has to be provided in the following form:
+
+<ul style='list-style: none;'>
+  <li><big>$\frac{dy(t)}{dt} = f(t,y(t))$</big></li>
+</ul>
+
+with an initial value
+
+<ul style='list-style: none;'>
+  <li><big>$y(t_0) = y_0$</big></li>
+</ul>
+
+To get a numeric solution, we replace the derivative on the LHS with a finite difference approximation:
+
+<ul style='list-style: none;'>
+  <li><big>$\frac{dy(t)}{dt}  \approx \frac{y(t+h)-y(t)}{h}$</big></li>
+</ul>
+
+then solve for $y(t+h)$:
+
+<ul style='list-style: none;'>
+  <li><big>$y(t+h) \approx y(t) + h \, \frac{dy(t)}{dt}$</big></li>
+</ul>
+
+which is the same as
+
+<ul style='list-style: none;'>
+  <li><big>$y(t+h) \approx y(t) + h \, f(t,y(t))$</big></li>
+</ul>
+
+The iterative solution rule is then:
+
+<ul style='list-style: none;'>
+  <li><big>$y_{n+1} = y_n + h \, f(t_n, y_n)$</big></li>
+</ul>
+
+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 $T_R$:
+
+<ul style='list-style: none;'>
+  <li><big>$\frac{dT(t)}{dt} = -k \, \Delta T$</big></li>
+</ul>
+
+or
+
+<ul style='list-style: none;'>
+  <li><big>$\frac{dT(t)}{dt} = -k \, (T(t) - T_R)$</big></li>
+</ul>
+
+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
+
+<ul style='list-style: none;'>
+  <li><big>$T(t) = T_R + (T_0 - T_R) \; e^{-k t}$</big></li>
+</ul>
+
+# --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:
+
+<ul>
+  <li><code>2 s</code></li>
+  <li><code>5 s</code> and</li>
+  <li><code>10 s</code></li>
+</ul>
+
+and compare with the analytical solution.
+
+**Initial values:**
+
+<ul>
+  <li>initial temperature <big>$T_0$</big> shall be <code>100 °C</code></li>
+  <li>room temperature <big>$T_R$</big> shall be <code>20 °C</code></li>
+  <li>cooling constant <big>$k$</big> shall be <code>0.07</code></li>
+  <li>time interval to calculate shall be from <code>0 s</code> to <code>100 s</code></li>
+</ul>
+
+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;
+}
+```