Calculator Input
Enter an initial-value problem in the form y' = f(x, y). Use common functions such as sin, cos, exp, sqrt, abs, and log.
Example Data Table
Sample setup: y' = x + y, x0 = 0, y0 = 1, h = 0.1, exact solution y = 2ex - x - 1.
| Step | xn | Approx yn | Used slope | Exact y(xn) | Abs Error |
|---|---|---|---|---|---|
| 0 | 0.000000 | 1.000000 | — | 1.000000 | 0.000000 |
| 1 | 0.100000 | 1.100000 | 1.000000 | 1.110342 | 0.010342 |
| 2 | 0.200000 | 1.220000 | 1.200000 | 1.242806 | 0.022806 |
| 3 | 0.300000 | 1.362000 | 1.420000 | 1.399718 | 0.037718 |
| 4 | 0.400000 | 1.528200 | 1.662000 | 1.583649 | 0.055449 |
| 5 | 0.500000 | 1.721020 | 1.928200 | 1.797443 | 0.076423 |
Formula Used
Forward Euler update:
yn+1 = yn + h · f(xn, yn)
xn+1 = xn + h
The method estimates the next y-value by taking the current slope and moving forward by one step size h.
How to Use This Calculator
- Enter the differential equation as a function of x and y.
- Provide the initial point x0 and y0.
- Enter a non-zero step size h.
- Choose either a target x value or a direct number of steps.
- Optionally add the exact solution to compare approximation error.
- Select the number of decimals for cleaner reporting.
- Click the solve button to generate the result block above the form.
- Use the export buttons to save the table as CSV or PDF.
Frequently Asked Questions
1) What does the forward Euler method do?
It estimates solutions to initial-value differential equations by moving in small x-steps. Each new y-value equals the previous y plus step size multiplied by the current slope.
2) When is this method accurate enough?
Accuracy improves when the step size is small and the equation changes smoothly. Large steps can create noticeable drift, instability, or poor approximations.
3) What if target x does not match the step size?
The solver checks alignment automatically. If target x is not an exact multiple of h from x0, enter a matching target or provide the step count directly.
4) Which functions can I type into the equation box?
You can use x, y, sin, cos, tan, exp, sqrt, abs, log, round, floor, ceil, min, max, and standard arithmetic operators including exponent notation with ^.
5) Why should I enter an exact solution?
The exact solution field is optional. It helps compare Euler approximations against known values, so the table and plot can include absolute error information.
6) Why is the graph empty after submission?
Graphs appear only after a successful calculation. If your formula is invalid or returns undefined values, fix the expression, interval, or step size and submit again.
7) Can I use a negative step size?
Yes. A negative h moves backward along the x-axis. The entered target or step count must still match the chosen direction correctly.
8) Is forward Euler suitable for stiff equations?
Usually not. Stiff equations often require very small steps or more stable methods, such as backward Euler or higher-order solvers.