91Ó°ÊÓ

In solving systems of differential equations, representing them in matrix form simplifies analysis. The given system of equations is: \( x' = 2x + 4y + 2 \) \( y' = x + 2y + 3 \). To put these into matrix form, we encapsulate the equations into vectors and matrices. This is done by letting:

• \( X' = \begin{bmatrix} x' \ y' \end{bmatrix} \)
• \( A = \begin{bmatrix} 2 & 4 \ 1 & 2 \end{bmatrix} \)
• \( X = \begin{bmatrix} x \ y \end{bmatrix} \)
• \( B = \begin{bmatrix} 2 \ 3 \end{bmatrix} \)

The matrix equation, therefore, looks like:\[ X' = AX + B \]. This transformation allows us to utilize linear algebra techniques in solving the system, making computations more manageable.

To solve the system for specific solutions that satisfy the non-homogeneous parts, we find a particular solution. The method of undetermined coefficients gives us a way to guess this solution.

Given the constant terms in the system, we can assume that the particular solution is also a constant vector. So, let's represent it as:\( \begin{bmatrix} x_p \ y_p \end{bmatrix} = \begin{bmatrix} A \ B \end{bmatrix} \), where \(A\) and \(B\) are constants we need to find. To find these values, substitute the particular solution back into the matrix form equation:\[ 0 = A \begin{bmatrix} 2 & 4 \ 1 & 2 \end{bmatrix} \begin{bmatrix} A \ B \end{bmatrix} + \begin{bmatrix} 2 \ 3 \end{bmatrix} \]. Solving this will lead to a system of algebraic equations that give us the constants \( A \) and \( B \).

The homogeneous solution solves the part of the system without the non-homogeneous term. For any matrix form system, you find this by setting the constant term vector \(B\) to zero.

To solve for it, you need to find the eigenvalues and eigenvectors of the matrix \(A\). The eigenvalues \(\lambda\) are found by solving:\[\det(A - \lambda I) = 0\], where \(I\) is the identity matrix.

Once the eigenvalues are determined, substitute them back into \((A - \lambda I)v = 0\) to find each corresponding eigenvector. The homogeneous solution will be a combination of these eigenvector solutions, scaled by arbitrary constants, describing how the system behaves with zero input.

Initial conditions specify the system's state at \(t = 0\) and are essential for finding the complete solution of a system involving differential equations. For the problem given:

• \(x(0) = 1\)
• \(y(0) = -1\)

These conditions help obtain a specific solution from general solutions by determining the arbitrary constants in homogeneous solutions.

To apply these, combine both the homogeneous and the particular solutions. Substitute \(t = 0\) and these initial values into the combined equation to solve for any unknown constants. This way, you can determine how the system was initially displaced, guiding it to follow a specific trajectory over time. These calculations ensure the solution fits precisely with the initial given conditions, striking a balance between theoretical and practical application of mathematics.