ODE System Solver
Enter a first-order system using variables t and y1..yN. Supported functions: sin, cos, tan, asin, acos, atan, sqrt, abs, exp, ln, log, log10, floor, ceil. Constants: pi, e. Power operator: ^.
Example Data Table
This example models a harmonic oscillator using a two-equation system.
| Item | Value |
|---|---|
| N | 2 |
| y1' | y2 |
| y2' | -y1 |
| t0 | 0 |
| tEnd | 6.283185 |
| h | 0.1 |
| y1(t0) | 0 |
| y2(t0) | 1 |
| Suggested method | RK4 |
Formula Used
Let the system be y' = f(t, y) with y = (y1..yN). Each method advances from (t_k, y_k) to (t_{k+1}, y_{k+1}) using step h.
- Euler: y_{k+1} = y_k + h f(t_k, y_k)
- Heun: k1=f(t_k,y_k), k2=f(t_k+h, y_k+h k1), then y_{k+1}=y_k+(h/2)(k1+k2)
- RK4: k1=f(t_k,y_k), k2=f(t_k+h/2, y_k+h k1/2), k3=f(t_k+h/2, y_k+h k2/2), k4=f(t_k+h, y_k+h k3), then y_{k+1}=y_k+(h/6)(k1+2k2+2k3+k4)
How to Use This Calculator
- Select how many equations your system contains.
- Enter each derivative expression using t and y1..yN.
- Provide initial values for all states at the start time.
- Set the end time and a step size that fits your dynamics.
- Pick a method and adjust sampling to control table size.
- Press Solve to view results above the form.
- Use CSV or PDF buttons to export the solution table.
FAQs
1) What kind of systems can I solve here?
Any first-order system expressed as y1'..yN' with numeric initial values. If you have higher-order equations, convert them into a first-order system by introducing extra state variables.
2) Which method should I choose?
Euler is fast but least accurate. Heun improves accuracy with modest cost. RK4 is usually the best balance of accuracy and stability for many smooth problems at practical step sizes.
3) Why do I get NaN or Infinity errors?
This typically means the system is stiff, the step size is too large, or the expression contains a division by zero or invalid operation. Reduce step size, verify expressions, and consider reformulating the model.
4) How do I write the equations correctly?
Use variables t and y1..yN exactly. Use operators +, -, *, /, ^ and supported functions like sin, exp, and sqrt. For example: y2 - 0.2*y1.
5) Can I solve stiff ODE systems accurately?
This tool uses explicit methods, which can struggle with stiff systems. You may need very small steps to remain stable. For stiff problems, implicit methods are often better, but they require more advanced solvers.
6) What does “sample every” mean?
It controls how often rows are recorded. A value of 1 records every step. A value of 10 records every tenth step plus the final point, reducing table size and export length.
7) Why is the final time exact even if the step does not fit?
The solver uses the chosen step size until near the end, then takes a smaller last step so the last row lands exactly on the requested end time.
8) Are the CSV and PDF exports identical?
The CSV contains all recorded rows. The PDF is a readable report and may truncate long tables, but it always includes method settings and key output structure.