Question

In each of problems 1-4 find the general solution of the given system of differential equations.

.
X = ( -1 2)
X
(-1 -1)

Answer 1

The student's question pertains to finding the general solution of a system of differential equations in matrix form. The solution involves finding eigenvalues and eigenvectors, and constructing the solution from these components.

Step-by-step explanation:

The student is asking for the general solution to a system of differential equations represented in matrix form as X = (-1 2) X (-1 -1). To find the general solution, one would typically follow a series of steps beginning from determining the eigenvalues and eigenvectors of the matrix, which would then be used to construct the general solution involving exponential functions of time.

The process includes:

1. Calculating the determinant of the matrix subtracted by λ (lambda) times the identity matrix.
2. Finding the eigenvalues by solving the characteristic equation.
3. Determining the corresponding eigenvectors for each eigenvalue.
4. Forming the general solution using the eigenvalues and eigenvectors.

Additionally, practicing substitution of the solutions into the original equation and differentiating them can be helpful for verification purposes.