Final answer:
The student is asked to find the general solution to a system of differential equations. This is done by finding eigenvalues and eigenvectors of the coefficient matrix and forming a linear combination of eigenvector solutions to construct the general solution.
Step-by-step explanation:
The question involves finding the general solution to a system of linear differential equations represented in matrix form. For example, a system like x₁′=x₁+2x₂ and x₂′=3x₁+2x₂ is equivalent to the matrix equation X′=AX, where A is the matrix of coefficients and X is the vector of variables. To solve, one needs to find the eigenvalues and eigenvectors of the matrix A. The general solution is a linear combination of the eigenvector solutions, typically represented as c₁e^{λ₁t}v₁ + c₂e^{λ₂t}v₂, where λ₁ and λ₂ are the eigenvalues, v₁ and v₂ are the corresponding eigenvectors, and c₁, c₂ are constants that can be determined from initial conditions.