Linear System ODE Solver Calculator

Calculator Inputs

General Settings

System dimension

Numerical method

Start time t₀

End time t₁

Time steps

Displayed decimals

Coefficient Matrix A

This matrix defines the linear coupling in x′ = Ax + b.

a11

a12

a13

a14

a21

a22

a23

a24

a31

a32

a33

a34

a41

a42

a43

a44

Constant Forcing Vector b

Use zeros for homogeneous systems or enter constants for nonhomogeneous forcing.

b1

b2

b3

b4

Initial State x(t₀)

x1(t₀)

x2(t₀)

x3(t₀)

x4(t₀)

Example Data Table

This sample uses x′ = Ax + b with A = [[0, 1], [-2, -3]], b = [0, 1], x(0) = [1, 0], t from 0 to 2, and RK4.

Step	t	x1	x2
0	0.000000	1.000000	0.000000
1	0.500000	0.919271	-0.231771
2	1.000000	0.797858	-0.227546
3	1.500000	0.697028	-0.170661
4	2.000000	0.625662	-0.115774

Formula Used

Model:
x′(t) = Ax(t) + b

Exact constant coefficient form:
x(t) = e^A(t-t₀)x(t₀) + ∫_t₀^t e^A(t-s)b ds

Forward Euler:
x_k+1 = x_k + h f(x_k), where f(x) = Ax + b

Heun predictor-corrector:
p = x_k + h f(x_k)
x_k+1 = x_k + (h / 2)[f(x_k) + f(p)]

Runge-Kutta 4:
k₁ = f(x_k)
k₂ = f(x_k + hk₁/2)
k₃ = f(x_k + hk₂/2)
k₄ = f(x_k + hk₃)
x_k+1 = x_k + h(k₁ + 2k₂ + 2k₃ + k₄)/6

How to Use This Calculator

Select the number of coupled equations in your system.
Enter the coefficient matrix A that multiplies the state vector x.
Enter the constant forcing vector b. Use zeros for homogeneous systems.
Enter the initial state values x(t₀), then choose the time interval and number of steps.
Pick a numerical method. RK4 is usually the most accurate among the included options.
Press Solve System to generate the summary, trajectory table, stability note, and equilibrium information.
Use the export buttons in the result section to save the computed table as CSV or PDF.
Review the component ranges and norm values to understand growth, decay, or oscillatory behavior.

FAQs

1. What kind of system does this calculator solve?

It solves constant coefficient linear systems of ordinary differential equations written as x′ = Ax + b. You can model up to four coupled first order equations with constant forcing.

2. Which method should I choose?

Use RK4 for the best accuracy in most cases. Heun offers a middle ground between speed and accuracy. Euler is fastest, but it can be much less accurate for stiff or rapidly changing systems.

3. Can it solve homogeneous and nonhomogeneous systems?

Yes. Enter zeros in the forcing vector b for a homogeneous system. Enter constant values in b to solve a nonhomogeneous linear system with constant input or forcing.

4. What does the stability note mean?

For two equations, the note is based on the eigenvalues of A. For larger systems, it uses Gershgorin discs as a fast stability check. That larger system note is helpful, but it remains heuristic.

5. Why might equilibrium be unavailable?

A unique equilibrium requires solving Ax + b = 0. If A is singular or nearly singular, the system may have no unique equilibrium, so the calculator reports that condition instead.

6. Does more time steps always improve accuracy?

Usually yes, because a smaller step size reduces numerical error. However, very large step counts increase computation and output size. A stable method with sensible steps is usually the best compromise.

7. What do the exported files contain?

The CSV export contains the computed time and state table. The PDF export captures the summary values, stability note, and tabulated solution so you can share or archive the results.

8. Is the displayed answer exact?

The table is a numerical approximation unless you independently evaluate the matrix exponential solution. The exact constant coefficient formula is shown so you can compare theory and computation.