Eigenvalues and Eigenvectors in Differential Equations

What is the coefficient matrix A for the given system of differential equations: x' = 6x + 5y and y' = x + 2y?

• [[6, 5], [1, 2]]

• [[6, 1], [5, 2]]

• [[x, y], [x', y']]

• [[6, 2], [5, 1]]

To find the eigenvalues (λ) of the matrix A = [[6, 5], [1, 2]], which determinant equation must be solved?

• det([[6+λ, 5], [1, 2+λ]]) = 0

• det([[6-λ, 5], [1, 2-λ]]) = 0

• det([[6, 5], [1, 2]]) = λ

• det([[λ-6, 5], [1, λ-2]]) = 0

What are the eigenvalues (λ) for the matrix A = [[6, 5], [1, 2]]?

• λ = 1, λ = 7

• λ = -1, λ = -7

• λ = 2, λ = 6

• λ = 0, λ = 8

When finding the eigenvector for the eigenvalue λ = 1, which matrix equation represents the system (A - λI)v = 0?

• [[5, 5], [1, 1]] * (a₁ a₂)' = (0 0)'

• [[7, 5], [1, 3]] * (a₁ a₂)' = (0 0)'

• [[6, 5], [1, 2]] * (a₁ a₂)' = (1 1)'

• [[6, 5], [1, 2]] * (a₁ a₂)' = (0 0)'

What is a possible eigenvector corresponding to the eigenvalue λ = 1, given the equation a1 + a2 = 0?

• (1, 1)

• (1, -1)

• (-1, -1)

• (0, 1)

For an eigenvalue λ = 7, the system of equations for the eigenvector components (a1, a2) simplifies to -a1 + 5a2 = 0. Which of the following is a valid eigenvector?

• (1, 5)

• (5, 1)

• (-5, 1)

• (1, -5)

If V1 and V2 are eigenvectors corresponding to eigenvalues λ1 and λ2 respectively, what is the general form of the solution X for a system of differential equations?

• X = c1*V1*e^(λ1*t) + c2*V2*e^(λ2*t)

• X = c1*V1*t + c2*V2*t

• X = c1*e^(λ1*t) + c2*e^(λ2*t)

• X = c1*V1 + c2*V2

In the general solution X = c1*V1*e^(λ1*t) + c2*V2*e^(λ2*t), what do λ1 and λ2 represent?

• Constants of integration

• Components of the eigenvectors

• Eigenvalues of the system's matrix

• Time-dependent variables