How to use this calculator
Type the coefficient matrix and, if you have one, the initial condition. The solver finds the eigenvalues and eigenvectors, writes the real general solution (including complex pairs and repeated eigenvalues), fits the constants, and classifies the equilibrium at the origin.
The eigenvalue method
Characteristic equationdet(A − λI) = 0
Distinct real λx = c1eλ₁tv1 + c2eλ₂tv2
Complex λ = α ± iβ, v = a + ibeαt(a cos βt − b sin βt) and eαt(a sin βt + b cos βt)
Repeated λ, one eigenvectoreλtv and eλt(tv + w), with (A − λI)w = v
Worked example
x′ = [1 1; 4 1]x, x(0) = (2, 0) (the default)
- det(A − λI) = (1 − λ)² − 4 = 0, so λ = 3 and λ = −1: a saddle point.
- Eigenvectors: (1, 2) for λ = 3 and (1, −2) for λ = −1.
- x(0) = c₁(1, 2) + c₂(1, −2) = (2, 0) gives c₁ = c₂ = 1, so x = e3t(1, 2) + e−t(1, −2).
Frequently asked questions
How do I tell the type of equilibrium?
For a 2 × 2 system: real eigenvalues of opposite sign give a saddle; same sign a node (stable if negative); complex eigenvalues a spiral (stable if the real part is negative) or a center when the real part is zero.