Revise Differential equations and slope fields for IB Mathematics: Analysis and Approaches (5.9)

A first-order differential equation has the form $\frac{dy}{dx}=f(x,y)$: it gives the gradient of a solution curve at any point $(x,y)$, but not the curve itself. You have already met the simplest kind — being told $\frac{dy}{dx}=2x$ and recovering $y=x^2+C$ by integrating. The new feature at HL is that the right-hand side may involve $y$ as well as $x$, as in $\frac{dy}{dx}=y-x$ or $\frac{dy}{dx}=\frac{2x}{y}$, and that we often build the equation ourselves from a described rate of change.

Because $\frac{dy}{dx}=f(x,y)$ hands you a gradient at every point, you can picture the solutions without solving anything. A slope (direction) field is built by choosing a grid of points and, at each one, drawing a short line segment whose gradient is $f(x,y)$. The result is a field of tangents that reveals the 'flow' of all solution curves. To draw a particular solution, start at an initial point and sketch a smooth curve that stays tangent to the segments as it moves.

Slope fields give a qualitative picture, but often we want an exact formula. That is possible for a special — and heavily examined — class of first-order equations: the separable ones. An equation is separable if it can be written as a function of $x$ times a function of $y$. The method is to gather all the $y$-terms (with $dy$) on one side and all the $x$-terms (with $dx$) on the other, and then integrate both sides.

$\text{If } \frac{dy}{dx}=g(x)h(y), \text{ then } \frac{1}{h(y)}\,dy=g(x)\,dx, \text{ so } \int\frac{1}{h(y)}\,dy=\int g(x)\,dx$.

Not every differential equation can be solved exactly — sometimes the integral has no elementary form. Euler's method sidesteps this by approximating the solution with a chain of short straight steps. Starting from the known point, you use the current gradient to step forward a small distance $h$ in $x$, update $y$, and repeat. Each step assumes the gradient is constant across the interval, which is why smaller steps give better accuracy.

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

The reason differential equations dominate applied questions is that so many real processes are defined by their rate of change. 'The rate of change is proportional to the amount present' is exactly $\frac{dN}{dt}=kN$, giving exponential growth ($k>0$) or decay ($k<0$) after separation. Mixing problems track an amount $Q$ in a tank where liquid flows in and out: the rate of change of $Q$ is the rate in minus the rate out, each rate being a concentration times a flow rate — again a first-order separable equation.

Paper 1 is marked analytically: each mark is tied to a specific line of working. An M mark rewards a correct method — the right approach, even if the arithmetic later slips. An A mark rewards accuracy and is DEPENDENT on the method: you cannot earn the A without the M it hangs off. The engine also accepts any correct equivalent form (so $y=\sqrt{2x^2+9}$ and $y^2=2x^2+9$ solved for $y$ score the same), applies ISW (ignore subsequent working once a correct answer appears) and allows FT/ECF follow-through on a genuine earlier slip. Study how each mark below is earned by one specific line.

Differential equations pull the whole calculus unit together: separation of variables is really integration in disguise, the modulus in $\ln|y|$ comes straight from your integration of $\frac{1}{y}$, and the exponential solutions reconnect with the growth and decay you met when studying $e^x$. Slope fields train the geometric habit of reading a derivative as a gradient — the same instinct behind tangents, stationary points and concavity — while Euler's method previews the numerical thinking used throughout applied mathematics. Master the three faces here and you have a template for modelling almost any process defined by how fast it changes.