Solver Inputs
This page solves the linear coupled system: dx/dt = ax + by + p and dy/dt = cx + dy + q.
Example Data Table
| Example | a | b | c | d | p | q | x0 | y0 | t0 | t end | Steps | Method | Expected Trend |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Stable decay example | -2 | 1 | -1 | -3 | 0 | 0 | 4 | 1 | 0 | 8 | 160 | RK4 | Both states move toward equilibrium. |
| Forced response example | -1.2 | 0.8 | -0.6 | -1.8 | 2 | -1 | 0 | 3 | 0 | 10 | 200 | Heun | States settle near a shifted equilibrium. |
Formula Used
System form
dx/dt = ax + by + p
dy/dt = cx + dy + q
Vector form: U′ = AU + K
Euler method
Un+1 = Un + h f(Un)
Heun method
Predictor: U* = Un + h f(Un)
Corrector: Un+1 = Un + (h/2)[f(Un) + f(U*)]
Runge-Kutta 4 method
Un+1 = Un + (h/6)(k1 + 2k2 + 2k3 + k4)
Equilibrium point
Set dx/dt = 0 and dy/dt = 0. When det(A) ≠ 0, the unique equilibrium is U* = -A-1K.
How to Use This Calculator
- Enter the four coupling coefficients a, b, c, and d.
- Provide optional constant forcing terms p and q.
- Set initial conditions for x(0) and y(0).
- Choose the simulation interval using start time and end time.
- Select the number of steps. Higher steps usually improve accuracy.
- Choose Euler, Heun, or Runge-Kutta 4.
- Click Solve System to show the results above the form.
- Review the summary cards, data table, time graph, and phase graph.
- Use the export buttons to save the computed table as CSV or PDF.
Frequently Asked Questions
1. What kind of systems does this solver handle?
It solves two first-order linear coupled differential equations with constant coefficients and optional constant forcing terms. This fits many modelling cases in maths, control, electronics, and simple interaction systems.
2. Which method should I choose?
Use RK4 for the best general accuracy, Heun for a balanced midpoint-style estimate, and Euler for quick rough checks. Smaller step sizes usually improve all three methods.
3. What do trace and determinant tell me?
They summarize the coefficient matrix. Together, they help classify the local behavior near equilibrium, such as stable node, spiral, saddle, or repeated-root behavior.
4. Why can the graph diverge quickly?
Rapid growth may be real system behavior, or it may indicate that the time step is too large. Try increasing the number of steps and compare methods.
5. What happens when the matrix is singular?
A singular coefficient matrix means the system may not have one unique equilibrium. The calculator still computes the numerical trajectory, but equilibrium reporting becomes limited.
6. What is the phase plot showing?
The phase plot traces y against x instead of time. It helps you see spirals, convergence, divergence, and directional flow more clearly.
7. Can I use negative time or coefficients?
Yes. Negative coefficients and mixed signs are allowed. Start time can also be negative, but the end time must still be greater than the start time.
8. Why export the table?
CSV is useful for spreadsheets and later analysis. PDF is useful for reports, assignments, and sharing the solver output with the chart summary.