Euler's Method Calculator
Solve an initial value problem dy/dx = f(x, y), y(x0) = y0 with the explicit (forward) Euler method — get a full step-by-step table of x, y and slope, an optional error column against a known exact solution, and a summary of the final and maximum error.
Input
The right-hand side of the differential equation, in terms of x and y.
If the exact solution is known, enter it here to add absolute/relative error columns.
Output
Summary
| Metric | Value |
|---|---|
| No data yet | |
Step-by-Step Table
| n | x | y (Euler) | f(x, y) | Exact y(x) | Absolute Error | Relative Error |
|---|---|---|---|---|---|---|
| No data yet | ||||||
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/eulers-method-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"functionExpr": "x + y",
"x0": "0",
"y0": "1",
"stepSize": "0.1",
"numSteps": "5",
"exactSolution": ""
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `eulers-method-calculator` tool (Euler's Method Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
<iframe
src="https://iotools.cloud/embed/eulers-method-calculator/"
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title="Euler's Method Calculator — iotools.cloud"
sandbox="allow-scripts allow-forms allow-same-origin allow-downloads allow-popups allow-popups-to-escape-sandbox"
allow="clipboard-write"
style="width:100%;border:1px solid #e5e7eb;border-radius:12px;overflow:hidden"></iframe>
<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|---|
| Need more credits? | View pricing |
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Guides
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
- Enter the right-hand side of the differential equation as f(x, y) — standard math notation works:
^for powers, plussqrt(),sin(),cos(),exp(),log(), and the other functions supported by mathjs's expression parser. - Enter the initial condition, x0 and y0.
- Set the step size (h) and the number of steps (n).
- Optionally enter the exact solution y(x), if you know it, to add absolute and relative error columns.
- 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.