Euler's Method Calculator

Solve any first-order ODE numerically with Euler's method. See the iteration table, the Euler polygon drawn on the slope field, and a convergence comparison at h, h/2 and h/4, with optional error analysis against a closed form.

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About Euler's Method Calculator

The Euler's Method Calculator numerically solves any first-order initial value problem of the form \( y' = f(x, y), \; y(x_0) = y_0 \) using the classical (forward) Euler method. It returns a full iteration table, plots the Euler polygon on top of a live slope field, compares the solution at three different step sizes so you can visually watch the method converge, and — if you supply the exact closed-form solution — produces a per-step error analysis.

What Is Euler's Method?

Euler's method is the simplest algorithm for approximating the solution of an initial value problem. Starting from a known point \( (x_0, y_0) \) on the solution curve, it repeatedly advances by a small step of size h along the local slope \( f(x, y) \):

y_n+1 = y_n + h · f(x_n, y_n), x_n+1 = x_n + h

Geometrically, each step is a short straight segment whose slope equals the differential equation's value at the current point. The resulting broken line — the Euler polygon — is an approximation to the true (usually curved) solution.

How Accurate Is It?

Euler's method is a first-order method. The local truncation error at each step is \( O(h^2) \) and the global error after integrating over a fixed interval is \( O(h) \). Practically:

Halving the step size roughly halves the global error.
The error grows linearly with the length of the integration interval.
Error is worst where the solution has high curvature.

The built-in step-size comparison (h, h/2, h/4) lets you see this linear convergence directly: enable the option and check that the three final values approach a common limit with each value roughly half as far from the limit as the previous one.

Reading the Chart

The visualization layers four kinds of information on a single coordinate plane:

Grey slope field — short line segments whose inclination equals \( f(x, y) \) at that point. Think of it as "the flow the ODE dictates". Any solution curve must be tangent to the field at every point.
Indigo Euler polygon — the stepwise numerical solution. Each segment starts at the previous grid point and points along \( f(x_n, y_n) \) for a distance of h.
Dashed green exact curve — present only when you supply the closed-form solution. The vertical orange dashed stubs are the signed local errors \( y_n - y_{\text{exact}}(x_n) \).
Orange and green comparison curves — the same problem re-run at h/2 and h/4, shown when step-size comparison is enabled.

How to Use This Calculator

Enter the right-hand side of the ODE in the field marked y' =. Use x and y as variables. Supported operators are + − × ÷ ^, and supported functions include sin, cos, tan, asin, acos, atan, exp, ln, log, log10, log2, sqrt, abs.
Set the initial conditions: the starting value x₀, the initial y₀ at that point, the step size h (positive to integrate forward, negative to integrate backward), and the number of steps n.
(Optional) Provide the exact solution y(x) if you know it. The calculator will compute \( |y_n - y(x_n)| \) at every step and report the maximum and final errors.
Toggle visualization options: the slope field is on by default; the step-size comparison overlays two extra curves at h/2 and h/4.
Click Run. The results section shows summary statistics, the chart, a convergence-comparison panel, and the full iteration table. Hovering a row highlights the corresponding point on the chart (and vice versa).

Worked Example

Consider \( y' = x + y, \; y(0) = 1 \) with h = 0.1 and 10 steps. The exact solution is \( y(x) = -x - 1 + 2e^x \). Applying Euler's formula gives:

n = 1: y₁ = 1 + 0.1 · (0 + 1) = 1.1 n = 2: y₂ = 1.1 + 0.1 · (0.1 + 1.1) = 1.22 n = 3: y₃ = 1.22 + 0.1 · (0.2 + 1.22) = 1.362 ⋮ n = 10: y₁₀ ≈ 3.1875, exact y(1) = 2e − 2 ≈ 3.4366

The final error is about 0.249. Halving h to 0.05 drops the final error to roughly 0.13, and halving again to 0.025 drops it to about 0.067 — clean linear convergence, exactly as the theory predicts.

Euler's Method vs Other Numerical Methods

Method	Order	Evaluations per step	Global error	Notes
Euler (forward)	1	1	O(h)	Simplest method; best for teaching and prototyping.
Improved Euler (Heun)	2	2	O(h²)	Averages slopes at start and end of step.
Midpoint (RK2)	2	2	O(h²)	Evaluates slope at the midpoint of each step.
Runge–Kutta 4 (RK4)	4	4	O(h⁴)	Workhorse general-purpose solver; very high accuracy per step.
Backward (implicit) Euler	1	1 (plus a root solve)	O(h)	Unconditionally stable; essential for stiff ODEs.

When Euler Goes Wrong

The forward Euler method can misbehave in three situations:

Step size too large — the polygon oscillates or diverges. The fix is to reduce h; the h, h/2, h/4 comparison makes this instantly visible.
Stiff ODEs — equations with rapidly decaying and slowly decaying modes simultaneously force h to be tiny for stability. Switch to an implicit (backward Euler) or BDF method.
Singularities in f(x, y) — division by zero, sqrt of a negative, or ln of a non-positive number will stop the integration. The calculator reports the offending step clearly.

Common Applications

Physics — Newton's second law as a first-order system, radioactive decay \( \dot{N} = -\lambda N \), Newton's law of cooling.
Biology & epidemiology — logistic growth \( \dot{y} = r\,y(1 - y/K) \), compartmental SIR models.
Economics — continuous compound interest, simple Solow growth models.
Chemistry — first-order reaction kinetics \( \dot{c} = -k c \).
Teaching — introducing the concept of numerical integration before moving on to RK4 or adaptive solvers.

Frequently Asked Questions

What is Euler's method?

Euler's method is the simplest numerical procedure for solving an initial-value problem y' = f(x, y), y(x0) = y0. At each step it advances the solution by y_{n+1} = y_n + h · f(x_n, y_n), effectively following the slope at the current point for a short distance h. It is first-order accurate, meaning the global error is O(h).

How accurate is Euler's method?

Euler's method has local truncation error O(h²) and global error O(h). Halving the step size roughly halves the global error. This is why the convergence comparison at h, h/2, and h/4 in this calculator is so instructive: you can see the error shrink approximately linearly with h.

When does Euler's method fail?

Euler's method can become unstable for stiff problems or when the step size is too large relative to the local curvature of the solution. You may see the numerical solution oscillate, blow up to infinity, or drift visibly from the true solution. Reducing h usually helps; for stiff equations, implicit methods such as Backward Euler are preferred.

How do I choose the step size?

Start with h that gives around 10 to 50 steps over the interval of interest. If the Euler polygon visibly deviates from the slope field or from your exact solution, halve h and rerun. Use the built-in h, h/2, h/4 comparison to check that the three curves are converging toward each other.

What is the difference between Euler's method and Runge-Kutta (RK4)?

Runge-Kutta fourth order evaluates the slope at four points per step and combines them with weights (1, 2, 2, 1)/6, giving global error O(h⁴) — several orders of magnitude better than Euler's O(h) for the same number of steps. Euler is still valuable for teaching the concept of numerical integration and for very simple or low-precision applications.

Can I use this for systems of ODEs?

This calculator handles a single scalar first-order ODE y' = f(x, y). For systems or for higher-order ODEs, you can rewrite the equation as a first-order system and use a dedicated system solver, or convert a second-order equation to two first-order ones and solve them component by component.

Can I integrate backward in time?

Yes — enter a negative step size h. The calculator will advance from x₀ in the negative direction for n steps. This is useful for reconstructing the past from a known present state.