What is the Euler method?

The Euler method is a first-order numerical procedure for solving ODEs with a given initial value. It advances the solution using the recurrence y_{n+1} = y_n + h * f(x_n, y_n), where h is the step size.

How accurate is the Euler method?

The local truncation error per step is O(h²) and the global error is O(h). Smaller step sizes give better accuracy but require more iterations. For stiff equations, consider higher-order methods like Runge-Kutta.

What expression syntax does this tool use?

Standard JavaScript math expressions: use x and y as variables, * for multiplication, ** for exponentiation, Math.sin(), Math.cos(), Math.exp(), Math.sqrt(), Math.log(), etc. Example: -2*x*y or Math.sin(x)+y.

Can I solve second-order ODEs?

Not directly. A second-order ODE must first be converted into a system of two first-order ODEs. This tool handles only a single first-order ODE y' = f(x, y).

Why does my solution blow up or give NaN?

Some ODEs diverge for large step sizes or specific initial conditions. Try a smaller step size h or a shorter interval. Check that your expression is correctly written and avoids division by zero.

How many steps can I compute?

The tool is capped at 500 steps to keep the browser responsive. For more steps, reduce the step size only if the interval is also shorter, or use dedicated ODE software.

Euler Method Calculator

Name: Euler Method Calculator
Availability: InStock
Author: Nham Vu

Numerically solve an ordinary differential equation using the Euler method, with a step-by-step table and chart.

Parameters

dy/dx = f(x, y)

Use x, y, Math.sin(), Math.exp(), etc.

x₀ (initial x)

y₀ (initial y)

x end

Step size h

Set parameters and click Solve to see the results.

Step-by-Step Table

Step	x	y	f(x, y) = dy/dx	Δy = h·f

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Summary

Numerically solve an ordinary differential equation using the Euler method, with a step-by-step table and chart.

How it works

Enter the ODE as a function of x and y — for example: x + y or -2*y.
Set the initial condition x0 and y0, then choose the step size h.
Set the target endpoint x_end where the approximation should stop.
Click Solve to run the Euler iterations and see the table of (x, y) values.
A line chart visualises the approximated solution curve.
Copy the table or individual rows for use in homework or reports.

Use cases

Check numerical ODE homework answers step by step.
Visualise how step size affects Euler method accuracy.
Prototype simple ODE solutions before implementing in code.
Teach or learn numerical methods with an interactive example.
Compare Euler approximations against known analytical solutions.
Generate quick solution tables for engineering or physics problems.

Frequently Asked Questions

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Last updated: 2026-05-23 · Reviewed by Nham Vu