Final answer:
The most general real-valued solution to a linear system of differential equations can be found using the method of eigenvalues and eigenvectors.
Step-by-step explanation:
The most general real-valued solution to a linear system of differential equations can be found using the method of eigenvalues and eigenvectors. We start by writing the system in matrix form, where A is the coefficient matrix, x is the vector of unknown variables, and b is the vector of constant terms:
Ax = b
To find the general solution, we first calculate the eigenvalues of A by solving the characteristic equation: det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix. Next, we find the corresponding eigenvectors for each eigenvalue. Finally, we use the eigenvalues and eigenvectors to construct the general solution:
x = c₁v₁e^(λ₁t) + c₂v₂e^(λ₂t) + ... + cₙvₙe^(λₙt)
where c₁, c₂, ..., cₙ are constants and v₁, v₂, ..., vₙ are the eigenvectors.