Matrix System Calculator
Solve x' = A x + b·g(t) for two or three linked equations. Enter matrix coefficients, forcing values, and time settings.
Example Data Table
This example models a damped forced second order system after conversion into matrix form.
| Item | Value | Meaning |
|---|---|---|
| A | [[0, 1], [-2, -3]] | Coupling matrix |
| x(t₀) | [1, 0] | Initial state |
| b | [0, 1] | Forcing direction |
| g(t) | 1 | Constant input |
| Range | 0 to 5, h = 0.05 | Numerical interval |
Formula Used
The calculator solves a non homogeneous linear system in matrix form.
The exact variation of constants form is:
For practical input types, this page uses fourth order Runge Kutta:
How To Use This Calculator
- Select a 2 by 2 or 3 by 3 matrix system.
- Enter every active coefficient in matrix A.
- Enter the initial vector and forcing vector.
- Choose the forcing function and its parameter.
- Set start time, target time, and step size.
- Press calculate to view results above the form.
- Use CSV or PDF buttons to save your report.
Matrix View of Forced Systems
A non homogeneous linear differential equation can describe coupled motion, current flow, population movement, or control response. The matrix form writes every linked variable in one compact model. It uses x prime equals A x plus f t. The matrix A controls internal coupling. The vector f t adds external input.
Why This Calculator Helps
Manual solving can become long when two or three equations interact. Eigenvalues may be repeated. Forcing may change with time. This page gives a practical numerical path. It accepts a square matrix, an initial vector, a time range, and a forcing rule. It then builds a step table and plots each component.
Numerical Method
The calculator uses fourth order Runge Kutta. This method estimates the slope at four points inside each step. It blends those slopes into a stable update. Smaller step sizes usually give better accuracy. Larger sizes run faster but can miss sharp behavior.
Reading The Results
The final vector shows the solution at the target time. The component chart shows growth, decay, oscillation, or steady behavior. The norm helps you judge overall size. The forcing preview shows the input applied during the interval. The CSV export supports spreadsheet checks. The PDF export creates a simple report for notes.
Good Input Practice
Start with a small time step. Compare it with a smaller step. If the answers barely change, the result is likely stable. Keep units consistent across all coefficients. Matrix entries should match the time unit. Initial values should use the same variable units. For exponential forcing, use a moderate rate first. For trigonometric forcing, choose an angular frequency that matches your model.
Study Uses
This tool is useful for systems courses, engineering mathematics, physics models, and applied linear algebra. It does not replace symbolic proof. It provides fast insight before deeper analysis. Use it to explore behavior, test examples, and prepare clean solution tables.
Extra Modeling Notes
Stable systems often move toward bounded paths. Unstable systems can grow quickly under small inputs. Oscillatory systems may cross zero many times. Always inspect the graph together with the table. A single final value can hide important transient behavior.
FAQs
What system does this calculator solve?
It solves x' = A x + b g(t), where A is a square matrix, x is the unknown vector, and b g(t) is the non homogeneous forcing term.
Can it solve symbolic differential equations?
No. It gives a numerical solution table and chart. Use symbolic software when you need an exact closed form with eigenvectors and integrals.
What does the matrix A represent?
The matrix A controls coupling between variables. Each row defines how current state components affect the derivative of one component.
How should I choose the step size?
Start small, such as 0.05. Then try half that value. If the final vector changes little, your step is probably reasonable.
What forcing functions are supported?
You can choose none, constant, exponential, sine, cosine, or polynomial forcing. The vector b controls the direction and size of input.
Why is the equilibrium sometimes unavailable?
An equilibrium estimate is shown for constant forcing when the matrix can be inverted numerically. Singular matrices may not have one unique equilibrium.
Can I export the full step table?
Yes. Use the CSV button for the complete computed table. Use the PDF button for a readable summary report.
Why does the graph grow very fast?
Fast growth can come from unstable eigenvalue behavior, large forcing, or a wide time range. Reduce the range and step size to inspect it.