AP Calculus BC

Euler's Method vs Exact Solution

Q: How do I tell whether Euler under- or overestimates?

Differentiate the differential equation to get \frac{d^2y}{dx^2} and check its sign across the whole interval you are stepping over. Concave up throughout means the estimate is too small, concave down throughout means it is too large, and a sign change means neither conclusion holds.

Use Euler's method when the question hands you a step size, or when the equation will not separate. Solve exactly by separating variables whenever the equation allows it. Euler underestimates a solution that stays concave up and overestimates one that stays concave down, because every step follows a tangent line.

Euler's method

Use when: An initial condition and a step size are given, or the equation cannot be separated and only an estimate is wanted.

Exact solution

Use when: The equation separates and the question asks for the particular solution $y = f(x)$ , or for a value with no approximation language.

Side by side

	Euler's method	Exact solution
What it produces	An approximate value at one input	A formula valid at every input
How it works	$y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x$	Separate, integrate both sides, then solve for the constant
Error	Present at every step and it accumulates	None
Direction of error	Under if the solution is concave up on the whole stepping interval, over if it is concave down there; if the concavity changes sign, neither conclusion holds	Not applicable
Works when the equation will not separate	Yes	No
Common trap	Reusing the starting slope for every step	Applying the initial condition before adding the constant

Euler's method is the exact solution viewed at low resolution. Over each step it throws away the curve and keeps the tangent line, so the estimate drifts a little further from the truth every time you step. Halving $\Delta x$ shrinks that drift but never removes it.

\frac{dy}{dx} = f(x, y) \quad \Longrightarrow \quad y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x

The error has a direction you can predict without computing anything, provided the concavity keeps one sign. Tangent lines sit below a concave up curve, so each Euler step lands low and the approximation comes out too small. For a concave down solution the tangents sit above and Euler comes out too large. Differentiating the differential equation gives $\frac{d^2y}{dx^2}$ , and its sign settles the over or under question when that sign holds across the whole interval you are stepping over. If the concavity changes sign inside the interval, the step errors point in opposite directions and neither conclusion holds.

The slope changes at every step

Each new slope is computed at the point you just landed on, not at the starting point. Reusing the first slope produces a straight line, which is the single most common way an Euler table goes wrong.

Frequently asked questions

Is Euler's method ever exact?

Only when the solution is a straight line over the interval, since the method follows tangent lines. Curvature over the interval produces error at every step, so an exact answer only happens by accident when errors of opposite sign cancel.

How do I tell whether Euler under- or overestimates?

Differentiate the differential equation to get $\frac{d^2y}{dx^2}$ and check its sign across the whole interval you are stepping over. Concave up throughout means the estimate is too small, concave down throughout means it is too large, and a sign change means neither conclusion holds.

Do I need the exact solution to run Euler's method?

No, and that is the point of it. Euler needs only the derivative formula and a starting point, so it works on equations that cannot be solved in closed form.

In the CED: Unit 7: Differential Equations