Differential Equation System Calculator
Analyze linear differential equation systems with eigenvalue analysis
Table of Contents
How to Use
- Enter the coefficient a₁₁ for the first equation's x term
- Enter the coefficient a₁₂ for the first equation's y term
- Enter the coefficient a₂₁ for the second equation's x term
- Enter the coefficient a₂₂ for the second equation's y term
- Click Calculate to see eigenvalues, eigenvectors, and stability analysis
What Are Differential Equation Systems?
A system of linear differential equations describes how multiple variables change over time in relation to each other. The 2×2 system has the form: dx/dt = a₁₁x + a₁₂y and dy/dt = a₂₁x + a₂₂y.
These systems appear in physics (coupled oscillators), biology (predator-prey models), economics (supply-demand dynamics), and engineering (control systems).
Eigenvalue Analysis
Eigenvalues determine the behavior of solutions. They are found by solving det(A - λI) = 0, which gives λ² - (trace)λ + (determinant) = 0.
| Eigenvalues | Classification | Behavior |
|---|---|---|
| Real, both negative | Stable node | Solutions approach origin |
| Real, both positive | Unstable node | Solutions move away from origin |
| Real, opposite signs | Saddle point | Unstable with stable/unstable directions |
| Complex with negative real part | Stable spiral | Spiral inward to origin |
| Complex with positive real part | Unstable spiral | Spiral outward from origin |
| Pure imaginary | Center | Closed orbits around origin |
Stability Criteria
The stability of the equilibrium at the origin depends on the trace and determinant:
- If det < 0: saddle point (unstable)
- If det > 0 and tr < 0: stable (node or spiral)
- If det > 0 and tr > 0: unstable (node or spiral)
- If det > 0 and tr = 0: center (neutrally stable)
- The discriminant tr² - 4det determines if eigenvalues are real or complex
Real-World Applications
- Population dynamics: predator-prey interactions (Lotka-Volterra equations)
- Mechanical systems: coupled springs and pendulums
- Electrical circuits: RLC circuits with multiple loops
- Chemical reactions: reaction kinetics with multiple species
- Economics: market equilibrium and price dynamics
- Control theory: feedback systems and stability analysis
Frequently Asked Questions
- What do eigenvalues tell us about the system?
- Eigenvalues determine how solutions evolve over time. Real eigenvalues indicate exponential growth or decay, while complex eigenvalues indicate oscillatory behavior. The sign of the real part determines stability.
- What are eigenvectors used for?
- Eigenvectors show the directions along which solutions move. They form the basis for the general solution and help visualize the phase portrait of the system.
- What does 'stable' mean in this context?
- A stable system means that solutions starting near the equilibrium point (origin) will approach it as time increases. An unstable system means solutions move away from the equilibrium.
- Can this calculator handle non-linear systems?
- No, this calculator is designed for linear systems only. Non-linear systems require linearization around equilibrium points before this analysis can be applied.