Euler Method Calculator
Euler’s method steps along the tangent: y′ = x + y with y(0) = 1 and h = 0.1 gives y(0.5) ≈ 1.72102 (exact 1.797443).
Use x and y, e.g. x + y, y − x^2 + 1, −2xy.
1.72102
Full answer, then cut to fewer decimals (not rounded):
- 1.72102
- 1.7210
- 1.721
- 1.72
- 1.7
- Steps
- 5
- Method order (global error)
- O(h)
| n | xₙ | yₙ |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 0.1 | 1.1 |
| 2 | 0.2 | 1.22 |
| 3 | 0.3 | 1.362 |
| 4 | 0.4 | 1.5282 |
| 5 | 0.5 | 1.72102 |
Step-by-step solution
- yₙ₊₁ = yₙ + h·f(xₙ, yₙ)
- x₀ = 0, y₀ = 1, h = 0.1, 5 steps
- y(0.5) ≈ 1.72102
Euler Method Formula
Euler: yₙ₊₁ = yₙ + h·f(xₙ, yₙ)
RK4: yₙ₊₁ = yₙ + h/6 (k₁ + 2k₂ + 2k₃ + k₄)
Worked Example
y′ = x + y, y(0) = 1, h = 0.1, up to x = 0.5:
- yₙ₊₁ = yₙ + h·f(xₙ, yₙ)
- x₀ = 0, y₀ = 1, h = 0.1, 5 steps
- y(0.5) ≈ 1.72102
Method order (global error): O(h)
| n | xₙ | yₙ |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 0.1 | 1.1 |
| 2 | 0.2 | 1.22 |
| 3 | 0.3 | 1.362 |
| 4 | 0.4 | 1.5282 |
| 5 | 0.5 | 1.72102 |
GATE tip
GATE asks for one or two Euler steps by hand. The first row of the table is y₁ = y₀ + h·f(x₀, y₀); compare methods by switching the dropdown.
More Numerical Methods Calculators
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Numerical Integration Calculator
Simpson: h/3 [y₀ + 4(y₁ + y₃ + …) + 2(y₂ + y₄ + …) + yₙ]
trapezoidal: h/2 [y₀ + 2(y₁ + … + yₙ₋₁) + yₙ]
Example: ∫₀¹ 1/(1 + x²) dx with n = 6: Simpson’s 1/3 rule ≈ 0.785397945234.
-
Newton-Raphson Method Calculator
xₙ₊₁ = xₙ − f(xₙ) ÷ f′(xₙ)
quadratic convergence near a simple root
Example: f(x) = x³ − 2x − 5 starting at x₀ = 2: Root x ≈ 2.09455148154.
-
Bisection Method Calculator
c = (a + b)/2; regula falsi c = (a·f(b) − b·f(a)) ÷ (f(b) − f(a))
keep the half where f changes sign
Example: f(x) = x³ − x − 2 on [1, 2]: Root x ≈ 1.5213804245.
-
Secant Method Calculator
xₙ₊₁ = xₙ − f(xₙ)(xₙ − xₙ₋₁) ÷ (f(xₙ) − f(xₙ₋₁))
order of convergence ≈ 1.618
Example: f(x) = x² − 612 starting from 10 and 30: Root x ≈ 24.7386337537.
Frequently asked questions
How accurate is Euler’s method compared with RK4?
Euler’s global error is proportional to h; RK4’s to h⁴. At h = 0.1 on y′ = x + y, Euler is off by about 0.08 while RK4 is off by about 10⁻⁶.
What is the modified Euler (Heun) method?
A predictor–corrector: predict with Euler, then average the slopes at both ends of the step. It is second-order accurate.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Euler Method Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/euler-method-calculator/
IEEE
Scientific Converters, "Euler Method Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/euler-method-calculator/