Final answer:
The solution to the given system of linear differential equations involves finding the eigenvalues and eigenvectors of the matrix and then expressing the solution as a combination of these components, including exponential functions of time.
Step-by-step explanation:
The question involves finding the general solution of a system of linear differential equations represented in matrix form. The system can be expressed as '
x' = Ax, where A is the matrix [ 12 -8; 16 -4 ] and x
is a vector of unknown functions. The general solution of this system involves determining the eigenvalues and eigenvectors of the matrix A and then expressing the solution in terms of these components, typically involving exponential functions of time multiplied by the eigenvectors. This process includes finding the characteristic polynomial, solving for eigenvalues, computing the corresponding eigenvectors, and then writing the general solution as a linear combination of the eigenvectors, with time-dependent coefficients.
Without the specifics of the differential equations, we use the method outlined above to find a solution set to such problems, applying concepts from linear algebra and differential equations.