Revise Differential equations: Euler's method and slope fields for IB Mathematics: Applications and Interpretation (5.8)

A first-order differential equation $\frac{dy}{dx}=f(x,y)$ is a statement about a gradient: it tells you how fast $y$ changes at every point. Setting one up is a translation exercise. 'The rate of growth is proportional to the population' becomes $\frac{dP}{dt}=kP$. 'A body cools at a rate proportional to how far it is above room temperature' becomes $\frac{d\theta}{dt}=-k(\theta-\theta_s)$. The skill is turning words about a rate into an equation — and then, later, turning the solution back into words with units.

A slope field draws the equation as a map. At a grid of points $(x,y)$ a short segment is drawn with gradient $f(x,y)$; together these segments show the direction every solution curve must follow. To sketch a particular solution, place your pencil at the initial condition and draw a smooth curve that stays tangent to the segments it passes through — never cutting across them. An isocline is a curve where the gradient is constant: setting $f(x,y)=m$ gives all points where solution curves have slope $m$, which helps you read the field.

Euler's method approximates a solution by taking short straight steps in the current gradient direction. From a known point it uses the tangent to jump a small distance $h$ along $x$, lands at a new point, recomputes the gradient there, and repeats. Each step is slightly off because the real solution curves, but with a small enough step length the chain of straight segments tracks the true curve closely.

Given $\frac{dy}{dx}=f(x,y)$ with initial condition $(x_0,y_0)$ and step length $h$:

$y_{n+1}=y_n+h\,f(x_n,y_n)$

$x_{n+1}=x_n+h$

Evaluate the gradient at the CURRENT point $(x_n,y_n)$, then update BOTH coordinates.

Some equations can be solved exactly. A first-order equation is separable if it can be written $\frac{dy}{dx}=g(y)\,h(x)$ — the $x$-part and the $y$-part multiply. Then you gather all the $y$'s on one side and all the $x$'s on the other and integrate: $\int\frac{1}{g(y)}\,dy=\int h(x)\,dx$. Integrating produces a single constant $C$; substitute the initial condition to pin it down and get the particular solution.

If $\frac{dy}{dx}=g(y)\,h(x)$, then

$\displaystyle\int \frac{1}{g(y)}\,dy=\int h(x)\,dx + C.$

Integrate both sides, then substitute the initial condition $(x_0,y_0)$ to find $C$.

The models you meet are all first-order equations solved the same way. Unlimited growth or decay uses $\frac{dy}{dt}=ky$, whose separable solution $y=y_0e^{kt}$ grows when $k>0$ and decays when $k<0$. Newton's law of cooling, $\frac{d\theta}{dt}=-k(\theta-\theta_s)$, says a body's temperature $\theta$ falls at a rate proportional to how far it sits above the surroundings $\theta_s$; the solution approaches $\theta_s$ without ever quite reaching it. In every case, set up the equation from the words, solve it (separation, or Euler if asked), and interpret the answer with units.

In an exam the marks are analytic: each is tied to a specific line of working. A method mark (M) rewards a correct approach even if the arithmetic later slips; an accuracy mark (A) rewards a correct result and is usually dependent on the method mark being earned. Follow-through (FT) means a correct step performed on your own earlier (wrong) value still scores, and equivalent correct forms are accepted. All of that protection exists only if your method is on the page. Study how each mark below is earned by a specific line.

Your GDC turns Euler's method into a recursion you can iterate quickly: store the step length, then repeatedly apply $y \to y+h\,f(x,y)$ and $x \to x+h$, reading off the value after the required number of steps — ideal for checking a hand table or for the many steps a Paper 2 question may need. For separable equations, use the graphing or solver features to confirm your particular solution passes through the initial condition and has the right shape. When you interpret a slope field, sanity-check by evaluating $f(x,y)$ at a couple of grid points and comparing with the drawn segments.

Everything here is one idea — a differential equation is a statement about a gradient — approached three ways: see the shape (slope field), estimate numerically (Euler), or solve exactly (separation). That toolkit powers the modelling questions that dominate Paper 2, where growth, decay and cooling recur constantly, and it connects straight back to integration, since solving a separable equation is just integration in disguise. Master the habit — set up the equation from the words, pick the technique the marks ask for, then interpret with units — and differential equations become routine.