Calculator Inputs
Example Data Table
Example inputs: f(x,y)=x+y, x₀=0, y₀=1, h=0.1, N=5 (Explicit Euler).
| Step | x | y | f(x,y) |
|---|---|---|---|
| 0 | 0.0 | 1.000000 | 1.000000 |
| 1 | 0.1 | 1.100000 | 1.200000 |
| 2 | 0.2 | 1.220000 | 1.420000 |
| 3 | 0.3 | 1.362000 | 1.662000 |
| 4 | 0.4 | 1.528200 | 1.928200 |
| 5 | 0.5 | 1.721020 | 2.221020 |
Your results may differ with a different method or rounding setting.
Formula Used
We solve the first‑order initial value problem: y' = f(x,y), with y(x₀)=y₀. Using step size h, we build points (xₙ, yₙ).
- Explicit Euler: yₙ₊₁ = yₙ + h·f(xₙ,yₙ), xₙ₊₁ = xₙ + h.
- Improved Euler (Heun): predict ŷ = yₙ + h·f(xₙ,yₙ), then yₙ₊₁ = yₙ + (h/2)[f(xₙ,yₙ)+f(xₙ₊₁,ŷ)].
- Midpoint Euler: k₁=f(xₙ,yₙ), k₂=f(xₙ+h/2, yₙ+(h/2)k₁), then yₙ₊₁ = yₙ + h·k₂.
How to Use This Calculator
- Type your derivative as f(x,y), like x+y or sin(x)-y.
- Enter x₀, y₀, and a positive step size h.
- Select a run mode: fixed steps N or a target x end.
- Optionally provide an exact y(x) to compute errors.
- Press Solve. Results appear above the form with a plot.
- Use Download CSV or Download PDF to export.
FAQs
1) What problems does Euler’s method solve?
It approximates solutions to first-order initial value problems like y′=f(x,y) with a starting point y(x₀)=y₀. It’s widely used for quick estimates and as a building block for more accurate methods.
2) Why do results change when I change step size?
Euler updates use straight-line steps, so error accumulates. A smaller h reduces the per-step error and usually improves accuracy, but increases the number of steps and computation time.
3) What is the difference between Euler and Improved Euler?
Improved Euler (Heun) uses a predictor and a correction by averaging slopes at the start and end of the step. This typically gives noticeably better accuracy than basic Euler for the same h.
4) When should I use the Midpoint variant?
Use Midpoint Euler when you want a simple upgrade that samples the slope at the middle of the step. It often performs well on smooth problems without requiring two full end-point evaluations.
5) What functions are supported in the input?
Common math functions such as sin, cos, tan, exp, log, log10, sqrt, abs, pow, min, and max are supported. Use pi and e as constants. Use x and y as variables.
6) Can I compute error against an exact solution?
Yes. Enter an exact y(x) expression in the optional field. The calculator then adds exact values and the difference y − exact at each step, making accuracy comparisons straightforward.
7) Why do I see “non‑finite” or math errors?
This happens when your expression causes invalid operations like dividing by zero, taking sqrt of a negative number, or overflow. Adjust the model, domain, or use a smaller h and shorter interval.
8) Are exported results identical to the table?
Yes. CSV is generated from the saved rows on the server. PDF is generated from the visible table. If you change rounding precision, export again to keep files consistent with the display.