Question

Find a fundamental matrix for the given system of equations.
x' = ( 2 -5 )
(1 -2 )

Answer 1

The fundamental matrix for the given system of equations is

Φ(t) = (
`e^(2t)`
`5e^((-2t))` )

( (1/5)
`e^{(2t)`
`4e^((-2t))` )

How can we find a fundamental matrix for the given system of equations?

we calculate the eigenvalues of the coefficient matrix

The characteristic equation of the matrix ( 2 -5 ) is:

(2 - λ)(-2 - λ) - (-5)(1) = λ² - 4

Solving for λ, we get λ = 2 and λ = -2.

Find eigenvectors corresponding to each eigenvalue:

For λ = 2:

Solving the system (A - 2I)x = 0: (0 -5)(x1) = (0) (1 -4)(x2) = (0)

We get the eigenvector v1 = (1, 1/5).

For λ = -2:

Solving the system (A + 2I)x = 0: (4 -5)(x1) = (0) (1 0)(x2) = (0)

We get the eigenvector v2 = (5, 4).

Constructing the fundamental matrix:

The fundamental matrix Φ(t) is formed by placing the eigenvectors as columns and multiplying by exponential terms:

Φ(t) = (
`e^(2t)`
`5e^((-2t))` )

( (1/5)
`e^{(2t)`
`4e^((-2t))` )

Therefore, the fundamental matrix for the given system of equations is:

Φ(t) = (
`e^(2t)`
`5e^((-2t))` )

( (1/5)
`e^{(2t)`
`4e^((-2t))` )