Topic 7.5 Approximating Solutions Using Euler's Method: approximate the solution of a differential equation at a point using Euler's method with a given step size and initial condition (BC).

What this topic is asking

The College Board (Topic 7.5, BC only) introduces Euler's method, a numerical way to approximate the solution of a differential equation when you cannot (or are not asked to) solve it exactly. Starting from a known point, you use the differential equation to read off the slope, then take a small straight-line step in that direction, repeating to march forward.

The update rule

Why it is just repeated tangent lines

Each Euler step is exactly the linear approximation (local linearization) from Unit 4: near $(x_n, y_n)$ the solution looks like its tangent line, whose slope is $f(x_n, y_n)$. Moving $\Delta x$ along that tangent changes $y$ by slope times $\Delta x$. The difference from a single linearization is that Euler re-reads the slope at each new point rather than using the original slope throughout, so the approximation bends to follow the slope field. This connection is worth stating on the exam: Euler's method is local linearization applied iteratively, with the slope field of Topics 7.3 and 7.4 giving the direction of each step.

A worked multi-step approximation

Over- and under-estimation from concavity

Whether Euler's method overshoots or undershoots the true solution depends on the concavity. Because each step follows a straight tangent line, when the actual solution curve is concave up the tangent lies below the curve, so Euler underestimates; when the solution is concave down, the tangent lies above, so Euler overestimates. You can judge concavity from the sign of $\frac{d^2y}{dx^2}$, found by differentiating the differential equation. This reasoning is a frequent free-response follow-up: after computing an Euler estimate, you may be asked whether it is greater or less than the true value, and the answer comes from the concavity of the solution, not from the size of the step.

Step size and accuracy

Euler's method is exact only in the limit of zero step size; with a finite $\Delta x$ it accumulates error at every step. Halving the step size roughly halves the error per step but doubles the number of steps, and overall the global error is proportional to $\Delta x$. The exam will specify the step size and the number of steps, so you rarely choose them yourself, but you should understand that more, smaller steps give a better approximation. The trade-off is more arithmetic, which is why exam problems use two or three steps with friendly numbers. Reading the requested step size and target carefully, and not stopping early or taking one step too many, is where most marks are won or lost.