Question

Solve the initial value problem x′ (t)=Ax(t) for t≥0, with x(0)=(3,1). Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by x′ = Ax. Find the directions of greatest attraction and/or repulsion. A=[ −1, -1]

​ [ 3,-5 ] Solve the initial value problem. x(t)=

Answer 1

The solution to the initial value problem involves calculating eigenvalues and eigenvectors of the matrix A and then constructing a solution involving exponentials of these eigenvalues.

Step-by-step explanation:

The given initial value problem is x'(t) = Ax(t) where A is a 2x2 matrix [−1, -1] [3, −5] and x(0) is the initial vector (3,1). To solve this system, we must first find the eigenvalues and eigenvectors of the matrix A. We do this by solving the characteristic equation
`det(A - λI) = 0` identity matrix. Once the eigenvalues are found, we find the corresponding eigenvectors. With these, we can write the general solution for the system in terms of exponentials raised to the eigenvalues and the associated eigenvectors.