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Euler Method Calculator

Forward Euler for a scalar first-order IVP y′=f(t,y). Same engine as Ordinary Differential Equation. Not RK4, not rk45, not a system. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 9/9 tests · Production surface contract 4/4 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
y(t1) of a scalar first-order IVP by forward Euler.
Scope
Scalar first-order only; method locked to euler
Verification
Engine tested · 9/9 tests · Production surface contract 4/4 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 golden · 2 boundary · 4 property · Production surface contract 4/4 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Stepy ← y + h f(t,y)
hh = (t1 − t0) / n
iFirst-order accurate. Linear rhs y′=k y has exact y0 e^{k(t1−t0)}. Polynomial rhs y′=p(t) has exact y0+∫p. RK4 lives on the sibling discovery URL.

How to use

1

Enter rhs, y0, t0, t1

Linear needs k. Poly needs coeffs. t1 must differ from t0.

2

Read y(t1) and |err|

The exact check is shown for linear and poly. RK4 is usually closer for the same n.

Example calculations

Common configurations with formula and result.

ϟ

y′=y, y(0)=1 at t=1

n=40, k=1

exact = e
2.65…
ϟ

y′=2t, y(0)=0 at t=1

poly coeffs 0,2

y = 1
1

Euler Method calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Forward Euler steps y ← y + h f(t,y). This page is method=euler on math.numerical.ode. Not a second ODE engine. Not RK4.
What it calculates
y(t1) of a scalar first-order IVP by forward Euler.
Inputs
  • rhs
  • y0
  • t0
  • t1
  • n?
  • k?
  • coeffs?
Outputs
  • y
  • exact
  • abs_err
  • h
  • n
Formula
y ← y + h f(t,y)
Assumptions
  • Scalar first-order only; method locked to euler
  • Not rk45, not systems, not PDE, not CAS
Units
  • dimensionless
Boundary conditions
  • t1 = t0 → INVALID_INPUT
  • n < 1 or n > 10000 → INVALID_INPUT
Example
euler linear k=1 y0=1 t0=0 t1=1 n=40 → exact=e
Validation cases

2 published on this page · 9/9 tests · Production surface contract 4/4 · View evidence

  • rhs=linear k=1 y0=1 t0=0 t1=1 n=40 → method=euler, exact=e
  • method=rk45 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

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Frequently asked questions

Key distinctions behind the calculation.

Is this a second ODE engine?

No. It is method=euler on math.numerical.ode, the same engine as /calc/math/ordinary-differential-equation.

Where does this run?

Locally in the browser by default. REST and MCP call the same ordinary-differential-equation engine.