Final answer:
The general solution of the differential equation system is obtained by finding the eigenvalues and eigenvectors of the coefficient matrix, forming a linear combination of eigenvectors and exponentials of the eigenvalues times time. The system is locally asymptotically stable because both eigenvalues are negative.
Step-by-step explanation:
To find the general solution for the system of differential equations given by dx/dt = −3x + 2y and dy/dt = 3x − 2y, we need to solve the system using methods for solving linear differential equations. First, we can write the system in matrix form as:
d/dt [x y]T = [ -3 2; 3 -2 ][x y]T,
where T denotes the transpose of the vector. To find the general solution, we find the eigenvalues and eigenvectors of the matrix [ -3 2; 3 -2 ]. The characteristic equation will be (-3 - λ)(-2 - λ) - (3×2) = λ2 + 5λ + 6 = 0, which yields the eigenvalues λ1 = -2 and λ2 = -3. The associated eigenvectors can then be found, and the general solution is a linear combination of the eigenvectors multiplied by the exponentials of the eigenvalues times t. To answer the stability question, if all eigenvalues of the matrix have negative real parts, then the system is locally asymptotically stable, which is the case here since both λ1 and λ2 are negative.