Solver Output
Enter coefficients, initial values, time range, and a numerical method. The solver will show the result above the form, plus tables, plots, and downloads.
Calculator Inputs
Formula Used
x′ = a11x + a12y + b1
y′ = a21x + a22y + b2
x(n+1) = x(n) + h·f1(x(n), y(n))
y(n+1) = y(n) + h·f2(x(n), y(n))
Predictor: use Euler once.
Corrector: average the initial slope and predicted slope.
The solver computes k1, k2, k3, and k4 slopes for both variables, then combines them with weights 1, 2, 2, and 1.
Set x′ = 0 and y′ = 0. Solve A·u + b = 0, where A = [[a11, a12], [a21, a22]] and u = [x, y]ᵀ.
trace(A) = a11 + a22
det(A) = a11a22 − a12a21
Eigenvalues follow from λ² − trace(A)λ + det(A) = 0.
How to Use This Calculator
- Enter the four matrix coefficients a11, a12, a21, and a22.
- Add forcing terms b1 and b2 if your system has constant inputs.
- Provide initial conditions x(t0) and y(t0).
- Choose start time, end time, and number of steps.
- Select Euler, Heun, or Runge-Kutta 4.
- Press Solve System to generate the table and graphs.
- Review equilibrium, eigenvalues, stability type, and final values.
- Download the current solution table as CSV or PDF.
Example Data Table
Example setup: x′ = y, y′ = −2x − 3y, x(0) = 1, y(0) = 0, t from 0 to 2, steps = 4, method = Runge-Kutta 4.
| Step | t | x(t) | y(t) | Interpretation |
|---|---|---|---|---|
| 0 | 0.00 | 1.0000 | 0.0000 | Initial state starts away from equilibrium. |
| 1 | 0.50 | 0.8450 | -0.4770 | Trajectory begins moving toward the origin. |
| 2 | 1.00 | 0.6004 | -0.4651 | Damping reduces the state amplitude. |
| 3 | 1.50 | 0.3966 | -0.3412 | System continues contracting toward equilibrium. |
| 4 | 2.00 | 0.2524 | -0.2258 | Stable decay remains visible in both states. |
FAQs
1. What kind of systems can this calculator solve?
This tool solves two coupled first-order linear differential equations with constant coefficients and optional constant forcing terms. It handles many introductory and intermediate systems used in mathematics, controls, and dynamics.
2. Which numerical method should I choose?
Euler is fastest but least accurate. Heun improves accuracy by correcting the slope. Runge-Kutta 4 is usually the best general choice when you want reliable results with moderate step counts.
3. Why do smaller step sizes matter?
Smaller steps usually reduce numerical error because the method updates the state more frequently. Very large steps may distort the trajectory, hide oscillations, or even create false instability.
4. What do the eigenvalues tell me?
Eigenvalues describe local behavior near equilibrium. Negative real parts imply decay, positive real parts imply growth, and imaginary parts indicate oscillatory motion. Their combination helps classify nodes, spirals, and saddle points.
5. What happens when det(A) equals zero?
If the determinant is zero, the matrix is singular. In that case, the system may not have a unique equilibrium point, so the calculator reports that a unique equilibrium cannot be guaranteed.
6. Is the displayed error an exact error?
No. The error shown is an estimate based on comparison with a finer Runge-Kutta 4 solution at the final time. It is useful for guidance, not a formal proof.
7. Can I use negative time or forcing values?
Yes. The calculator accepts negative coefficients, negative forcing terms, and any start time smaller than the end time. The equations are solved over the interval you specify.
8. What does the phase plot show?
The phase plot traces y against x instead of plotting both against time. It helps you see spirals, attraction toward equilibrium, divergence, and directional flow in state space.