欧拉法计算器

The Euler's Method Calculator numerically solves an initial value problem — a differential equation dy/dx = f(x, y) together with a starting point y(x0) = y0 — using the explicit (forward) Euler method. It steps forward from the starting point in fixed increments of size h, producing a full table of every intermediate x, y and slope, plus an optional error column when the true (exact) solution is known.

It's built for students checking a numerical-methods homework problem, and anyone who needs a quick approximate solution to an ODE without setting up a spreadsheet by hand.

How to use it

1. Enter the right-hand side of the differential equation as f(x, y) — standard math notation works: ^ for powers, plus sqrt(), sin(), cos(), exp(), log(), and the other functions supported by mathjs's expression parser.
2. Enter the initial condition, x0 and y0.
3. Set the step size (h) and the number of steps (n).
4. Optionally enter the exact solution y(x), if you know it, to add absolute and relative error columns.
5. Read the Summary table for the final approximation (and, if an exact solution was given, the final and maximum error), and the Step-by-Step Table for every intermediate value.

Results update automatically as you type. For example, dy/dx = x + y with y(0) = 1, h = 0.1, over 5 steps approximates y(0.5) ≈ 1.72102.

What is Euler's method?

Euler's method approximates the solution curve of dy/dx = f(x, y) by repeatedly taking a small step in the direction of the current slope:

x_(n+1) = x_n + h
y_(n+1) = y_n + h · f(x_n, y_n)

Starting from (x0, y0), each new point uses the slope f(x, y) computed at the previous point — that's what makes it "explicit" (or "forward") Euler, the simplest numerical method for solving an ordinary differential equation. It's a first-order method: halving the step size roughly halves the error, so a smaller h gives a more accurate approximation at the cost of more steps.

Why does the calculator ask for an exact solution?

If you already know (or have separately derived) the closed-form solution y(x) — for example because the ODE is separable or linear — entering it lets the calculator show exactly how far off the Euler approximation is at every step, via the Absolute Error (|exact − approximate|) and Relative Error (absolute error as a percentage of the exact value) columns, plus the final and maximum error in the summary. This is the standard way to sanity-check a numerical method: apply it to a problem you can also solve exactly, and see how the error grows.

What functions can I enter?

Anything mathjs's expression evaluator understands: polynomials (x^2 - y), trigonometric functions (sin(x), cos(y)), exponentials and logarithms (exp(x), log(y)), and combinations of these, in terms of x and y. The exact solution field is the same expression language, but in terms of x only.

To evaluate a single expression directly, see the Math Evaluator. For the slope of a secant line between two points on a function, see the Average Rate of Change Calculator.

Privacy

This calculator runs entirely in your browser. Your differential equation, initial condition, and every computed value are never uploaded, logged, or stored — the computation happens locally on your device.