Compute exact states, growth behavior, and numerical checks. Review eigenvalues, determinants, traces, and trajectories instantly. Download tidy outputs for lessons, reports, practice, and archiving.
| a11 | a12 | a21 | a22 | x1(0) | x2(0) | t | Steps | Example x(t) |
|---|---|---|---|---|---|---|---|---|
| 2 | -1 | 1 | 0 | 1 | 2 | 1 | 10 | [0, 2.718282] |
The calculator solves the system X′ = AX with X(0) = X0.
The exact solution is X(t) = eAtX0.
For a two by two matrix, the page uses trace and determinant.
Let τ = trace(A) / 2 and B = A - τI.
If τ² - det(A) is positive, the page uses hyperbolic terms.
If τ² - det(A) is zero, the repeated root form is used.
If τ² - det(A) is negative, the page uses sine and cosine terms.
The Euler check uses Xn+1 = Xn + hAXn.
This gives an easy numerical comparison at the same target time.
This calculator solves a two by two linear matrix differential equation. It handles systems written as X′ = AX with initial values. You enter the matrix, the starting vector, the target time, and the number of steps. The tool then returns an exact state estimate, derivative values, eigenvalue data, and a numerical step table. That mix helps students check both theory and computation in one place.
Matrix differential equations appear in mechanics, circuits, control models, population flow, and finance. Many real systems change together. A single equation cannot describe that interaction well. A matrix system keeps the links between variables visible. It also shows whether a model grows, decays, oscillates, or shifts direction over time. This makes the topic useful in both classroom work and applied problem solving.
The exact solution uses the matrix exponential for a two by two system. The page first finds the trace and determinant. It then uses them to study stability and eigenvalues. After that, it builds eAt and multiplies it by the initial vector. The calculator also creates an Euler method table. That numerical table helps you compare a step based estimate with the exact state at the selected time.
The output is not limited to one answer. You also see the transformed state, the derivative at the chosen time, the characteristic equation, and a simple behavior label. Those details help when you must explain your work. They are also helpful for revision because they connect formulas, interpretation, and numeric evidence. The example table and export tools make it easier to save results for notes, homework, and practice sets.
Start with a system from your textbook or lecture notes. Enter each matrix value carefully. Add the initial vector and choose a practical time. Use more steps when you want a smoother Euler table. After solving, compare the exact values with the numerical values. Then review the formula section to understand why the method works. This process builds confidence, improves checking habits, and supports faster exam preparation. It also suits tutoring and independent self study sessions.
It solves a two by two linear matrix differential equation with an initial vector. The system is written as X′ = AX and evaluated at a chosen time.
Yes. The page computes the exact state with a matrix exponential method. It also builds an Euler table so you can compare approximation and exact behavior.
Eigenvalues help describe system behavior. They show whether the solution grows, decays, oscillates, or forms a saddle pattern. That makes interpretation faster during study.
Yes. The calculator accepts negative or positive time. The Euler step size changes automatically because it is based on target time divided by steps.
Euler is an approximation method. Smaller step sizes usually improve accuracy. Exact and numerical values may differ more when the system changes quickly.
The label summarizes the phase behavior from the eigenvalue pattern. Common labels include stable node, unstable node, saddle point, stable spiral, and center type oscillation.
Enter the starting state values x1(0) and x2(0). These values define the position of the system at time zero before any evolution occurs.
Yes. After solving the system, use the CSV or PDF buttons. They export the summary values and the Euler method table for notes or reports.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.