Solver Inputs
Example Data Table
This example uses the default nonlinear predator-prey style system: dx/dt = x(1.1 - 0.4y), dy/dt = y(-0.7 + 0.25x), with t₀ = 0, x₀ = 2, y₀ = 1, h = 0.1, and RK4.
| Step | t | x(t) | y(t) | dx/dt | dy/dt | Norm |
|---|---|---|---|---|---|---|
| 0 | 0.000000 | 2.000000 | 1.000000 | 1.400000 | -0.200000 | 2.236068 |
| 1 | 0.100000 | 2.145818 | 0.981963 | 1.517555 | -0.160596 | 2.359828 |
| 2 | 0.200000 | 2.303747 | 0.967918 | 1.642187 | -0.120083 | 2.498823 |
| 3 | 0.300000 | 2.474486 | 0.958000 | 1.773711 | -0.077961 | 2.653459 |
| 4 | 0.400000 | 2.658706 | 0.952399 | 1.911717 | -0.033642 | 2.824142 |
| 5 | 0.500000 | 2.857022 | 0.951368 | 2.055493 | 0.013562 | 3.011258 |
| 6 | 0.600000 | 3.069960 | 0.955235 | 2.203944 | 0.064469 | 3.215141 |
Formula Used
This calculator solves a nonlinear first-order system of two coupled ordinary differential equations:
The implemented methods are:
xn+1 = xn + h·f₁(tn, xn, yn)
yn+1 = yn + h·f₂(tn, xn, yn)
Predictor:
x* = xn + h·f₁(tn, xn, yn)
y* = yn + h·f₂(tn, xn, yn)
Corrector:
xn+1 = xn + (h/2)[f₁(tn, xn, yn) + f₁(tn+h, x*, y*)]
yn+1 = yn + (h/2)[f₂(tn, xn, yn) + f₂(tn+h, x*, y*)]
k₁ = f(tn, sn)
k₂ = f(tn + h/2, sn + hk₁/2)
k₃ = f(tn + h/2, sn + hk₂/2)
k₄ = f(tn + h, sn + hk₃)
sn+1 = sn + (h/6)(k₁ + 2k₂ + 2k₃ + k₄)
Here, the state vector is s = [x, y], h is the step size, and the norm shown in the results table is √(x² + y²).
How to Use This Calculator
- Enter the first nonlinear equation for dx/dt using t, x, and y.
- Enter the second nonlinear equation for dy/dt in the same format.
- Provide initial time t₀ and initial states x₀, y₀.
- Choose the step size h and total number of steps.
- Select Euler, Heun, or RK4 depending on the balance you want between speed and accuracy.
- Set decimal places for the output table and a magnitude guard for unstable solutions.
- Press Solve System to generate summary metrics, solution tables, and Plotly graphs.
- Use the CSV or PDF buttons to export the calculated results for documentation, review, or sharing.
8 FAQs
1) What kind of systems can this solver handle?
It handles two coupled first-order nonlinear equations. Your expressions may include time, cross-coupling terms, powers, trigonometric functions, exponentials, and logarithms, provided the formulas remain numerically valid across the chosen interval.
2) Which method should I choose first?
RK4 is usually the best starting choice because it balances stability and accuracy well. Euler is fastest but rougher. Heun improves Euler by averaging slopes and often performs better with moderate step sizes.
3) Why does the solution explode or stop early?
That usually means the system is unstable for your chosen parameters, the step size is too large, or the equations hit invalid math regions such as division by zero or square roots of negative values.
4) Can I use time-dependent forcing terms?
Yes. Because the variable t is available in each expression, you can enter nonautonomous systems such as sin(t)-x*y or y+exp(-t) directly.
5) What does the norm value mean?
The norm is the Euclidean size of the state vector, computed as √(x² + y²). It is useful for spotting growth, decay, bounded oscillations, and general changes in overall trajectory magnitude.
6) How small should the step size be?
Start with a moderate step, then reduce it until the final values and trajectory change only slightly. Stiffer or rapidly varying systems usually need smaller steps for reliable results.
7) Does this solver provide an exact solution?
No. It provides numerical approximations. The quality depends on the chosen method, step size, and the system’s behavior. Comparing multiple step sizes is a practical way to assess convergence.
8) Can I export the full result table?
Yes. After solving, use the CSV button for spreadsheet work or the PDF button for reports. Both exports use the currently displayed iteration table and summary context.