Calculator Input
Enter a first-order initial value problem in the form y' = f(x,y).
Supported functions include sin, cos, tan, sqrt, log, exp, abs, and pow.
Example Data Table
This sample uses y' = x + y, y(0)=1, target x=0.6, and step h=0.2.
| Step | x | Euler y | Heun y | RK4 y |
|---|---|---|---|---|
| 1 | 0.2 | 1.2000 | 1.2400 | 1.2428 |
| 2 | 0.4 | 1.4800 | 1.5768 | 1.5836 |
| 3 | 0.6 | 1.8560 | 2.0257 | 2.0442 |
Formula Used
Euler Method
Euler method uses the starting slope to move one step forward.
The formula is y(n+1) = y(n) + h f(x(n), y(n)).
It is fast, but its accuracy depends strongly on a small step size.
Heun Improved Euler Method
Heun method first predicts a value with Euler.
Then it corrects the value using the average of two slopes.
The formula is y(n+1) = y(n) + h(k1 + k2)/2.
Fourth Order Runge-Kutta Method
RK4 uses four slope estimates in one step.
Its formula is y(n+1) = y(n) + h(k1 + 2k2 + 2k3 + k4)/6.
It gives strong accuracy for many smooth first-order equations.
How to Use This Calculator
- Enter the right side of the equation as
f(x,y). - Add the initial values for
xandy. - Enter the target x value.
- Choose the step size and numerical method.
- Add an exact solution if you want error comparison.
- Press the calculate button to show the result above the form.
- Download the solution table as CSV or PDF.
About Online Differential Equation Solving
Why Numerical Methods Matter
Differential equations describe changing systems. They appear in motion, growth, circuits, cooling, finance, and many other models. Some equations have clean symbolic answers. Many practical equations do not. A numerical calculator helps estimate the solution at chosen points. It turns a continuous problem into a clear step table.
Initial Value Problems
This calculator focuses on first-order initial value problems. You enter the slope rule as y prime equals f of x and y. You also enter one known point. The solver then moves forward or backward from that point. Each new value depends on the previous value and the selected method.
Choosing a Method
Euler method is simple and useful for quick estimates. It follows one slope per step. Heun method improves that estimate by averaging a starting slope and an ending slope. RK4 is usually more accurate. It samples four slopes inside each step. For smooth equations, RK4 often gives reliable results with fewer steps.
Step Size and Accuracy
Step size controls the distance between calculated points. A smaller step usually improves accuracy. It also creates more rows. Large steps are faster, but they may miss important curve behavior. When an exact solution is known, enter it in the optional field. The calculator then reports absolute error for each row.
Study and Reporting Use
The solution table is useful for homework checks, lesson notes, lab reports, and model comparison. It shows slopes, step values, predicted values, corrected values, and final output. CSV export helps spreadsheet review. PDF export gives a clean printable report. Always check the equation syntax and use realistic step sizes.
FAQs
1. What type of equation does this calculator solve?
It solves first-order initial value problems written as y prime equals f(x,y). It uses numerical methods instead of symbolic algebra.
2. Which method is most accurate?
RK4 is usually the most accurate option here. It uses four slope estimates per step, which improves stability for many smooth equations.
3. Can I calculate backward from the initial point?
Yes. Enter a target x value smaller than the initial x value. The calculator automatically applies the step direction backward.
4. What functions can I use in the equation?
You can use common safe functions such as sin, cos, tan, sqrt, log, log10, exp, abs, floor, ceil, and pow.
5. What does the exact solution field do?
It compares the numerical answer with a known y(x). If you leave it empty, the calculator skips error calculation.
6. Why does step size matter?
Step size controls accuracy and table length. Smaller steps often improve results, but they require more calculations and longer tables.
7. Can I export the answer?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a printable report.
8. Why do some equations fail?
An equation may fail because of invalid syntax, division by zero, unsupported functions, or values that become infinite during calculation.