Euler's Method Approximator

Math Calculus • Differential Equations

Written by STEM Calculators Team Published December 31, 2025 Updated February 24, 2026

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3. Euler's Method Approximator

Numerically approximate the IVP \(y' = f(x,y)\), \(y(x_0)=y_0\) using \(\;y_{n+1}=y_n+h\,f(x_n,y_n)\).

Differential equation

\(f(x,y)\): Exact \(y(x)\) (optional):

Use x and y in \(f\). Exact \(y(x)\) (if known) uses x only. Supports pi, e, sqrt(), sin, cos, exp, etc. Use * for multiplication.

Initial condition + step

\(x_0\): \(y_0=y(x_0)\): End \(x\) (target): Step size \(h\):

The tool chooses the integration direction based on \(x_{\text{end}}-x_0\) and uses \(|h|\). It computes \(N=\lceil |x_{\text{end}}-x_0|/|h| \rceil\) steps and reports the actual endpoint \(x_N=x_0+Nh\).

Graph settings (optional)

x min: x max: y min: y max: Max steps: Engine preference: Euler draw mode:

Wide outputs (tables/canvas/steps) scroll horizontally on small screens.

Ready

Enter \(f(x,y)\), \(x_0\), \(y_0\), \(h\), \(x_{\text{end}}\) and click “Compute”.

Iteration table

Graph

Euler path (pan/zoom) — axes/ticks shown

Drag to pan; mouse wheel to zoom; double-click resets. Axes are \(x\) and \(y\) in your input units.

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Theory — Euler’s Method (Forward Euler)

IVP setup

An initial value problem (IVP) has the form \[ y' = f(x,y), \qquad y(x_0)=y_0. \] The goal is to approximate \(y(x)\) on an interval starting from \(x_0\).

Euler update rule

Choose a step size \(h\) and define grid points \[ x_n = x_0 + nh. \] Euler’s method uses the tangent slope \(f(x_n,y_n)\) at \((x_n,y_n)\) to step forward: \[ y_{n+1} = y_n + h\,f(x_n,y_n). \] Geometrically, you move from \((x_n,y_n)\) to \((x_{n+1},y_{n+1})\) along a short line segment whose slope matches the differential equation at the left endpoint.

Worked example: \(y'=y\), \(y(0)=1\), \(h=0.1\) to \(x=1\)

Here \(f(x,y)=y\). Starting at \((x_0,y_0)=(0,1)\): \[ y_1 = y_0 + h\,y_0 = 1 + 0.1(1)=1.1 \] \[ y_2 = y_1 + h\,y_1 = 1.1 + 0.1(1.1)=1.21 \] Continuing for 10 steps (so \(x_{10}=1\)) gives \(y(1)\approx 2.5937\). The exact solution is \(y(x)=e^x\), so \(y(1)=e\approx 2.7183\), and Euler’s method underestimates here.

Accuracy and step size

Euler’s method has:

Local truncation error proportional to \(h^2\) (per step).
Global error proportional to \(h\) over a fixed interval (after many steps).

So halving \(h\) typically reduces the overall error by about a factor of 2 (when \(f\) is smooth and the solution stays well-behaved).

The calculator includes a “Refine \(h\)” button to quickly halve the step size and observe convergence.

University extension: compare with Runge–Kutta

Higher-order methods (like RK4) evaluate \(f(x,y)\) multiple times per step to reduce error substantially. Euler is a first-order method; RK4 is fourth-order and is usually dramatically more accurate for the same step size.

Systems (preview)

For a system \(\mathbf{y}'=\mathbf{f}(x,\mathbf{y})\), Euler generalizes to \[ \mathbf{y}_{n+1}=\mathbf{y}_n+h\,\mathbf{f}(x_n,\mathbf{y}_n), \] applying the same update to every component.

Frequently Asked Questions

What does Euler's method approximate?

Euler's method approximates the solution y(x) to an initial value problem y' = f(x,y) starting from (x0, y0). It advances in steps of size h using y_{n+1} = y_n + h f(x_n, y_n).

How do I choose a good step size h for Euler's method?

Smaller |h| typically improves accuracy but increases the number of steps and computation time. A practical approach is to halve h and check whether the table and graph change noticeably over the interval.

Why does the calculator sometimes step backward in x?

If the target x_end is less than x0, the method must move in the negative x direction. The tool determines the direction from x_end - x0 while using the magnitude |h| for the step size.

What is the difference between connect-points and stair-step drawing?

Connect-points draws straight line segments between successive Euler points (x_n, y_n). Stair-step draws a horizontal move in x followed by a vertical move in y, emphasizing the piecewise update behavior.