91Ó°ÊÓ

Verifying that a vector is a particular solution of a nonhomogeneous linear system is essential for understanding the system's behavior. Essentially, this verification confirms that the given vector satisfies the differential equation when plugged into it. In the context of the provided exercise, the vector function \( \mathbf{X}_{p}(t) = \begin{pmatrix} \sin 3t \ 0 \ \cos 3t \end{pmatrix} \) is alleged to be a particular solution. To prove this, we need to go through a process of

• differentiating the vector with respect to time \( t \), producing \( \mathbf{X}_{p}^{\prime}(t) \),
• multiplying the original vector \( \mathbf{X}_{p} \) by the system's matrix, and
• adding the nonhomogeneous part to the product.

Only when the result matches \( \mathbf{X}_{p}^{\prime}(t) \) can we confirm that the given particular solution truly solves the original differential equation.

Matrix-vector multiplication is a fundamental operation in linear algebra crucial for solving linear systems. In the exercise, we need to multiply matrix \( A \) with the vector \( \mathbf{X}_{p}(t) \). The matrix \( A \) is defined by:\[A = \begin{pmatrix} 1 & 2 & 3 \ -4 & 2 & 0 \ -6 & 1 & 0 \end{pmatrix}\]Multiplying this matrix with the vector\[\mathbf{X}_{p}(t) = \begin{pmatrix} \sin 3t \ 0 \ \cos 3t \end{pmatrix}\]involves working through each row of the matrix and performing the dot product with the vector. This process results in another vector that is part of verifying whether \( \mathbf{X}_{p}(t) \) satisfies the differential equation.

• The result of this multiplication is \( \begin{pmatrix} \sin 3t + 3 \cos 3t \ -4 \sin 3t \ -6 \sin 3t \end{pmatrix} \).

The accuracy of this step is vital for the correctness of the solution verification process.

Differentiation of vectors is analogous to differentiating scalar functions but is applied component-wise. To verify the particular solution \( \mathbf{X}_{p}(t) \), we differentiate each component of the vector:\[\mathbf{X}_{p}(t) = \begin{pmatrix} \sin 3t \ 0 \ \cos 3t \end{pmatrix}.\]Differentiating yields:\[\mathbf{X}_{p}^{\prime}(t) = \begin{pmatrix} 3 \cos 3t \ 0 \ -3 \sin 3t \end{pmatrix}.\]

• The derivative of \( \sin 3t \) is \( 3 \cos 3t \) and the derivative of \( \cos 3t \) is \(-3 \sin 3t \).
• The differentiation provides the instantaneous rate of change of the vector components concerning time \( t \).

This differentiated vector is then compared against the combination of matrix-vector multiplication results and the nonhomogeneous part to establish solution consistency.

Solution consistency involves ensuring that every part of the vector equation balances correctly. After differentiating the vector and performing matrix-vector multiplication, the results need to be summed with the nonhomogeneous part of the equation:\[\mathbf{b}(t) = \begin{pmatrix} -\sin 3t \ 4 \sin 3t \ 3 \sin 3t \end{pmatrix}.\]Combine this with the earlier computed result\[A \mathbf{X}_{p}(t) + \mathbf{b}(t) = \begin{pmatrix} \sin 3t + 3 \cos 3t \ -4 \sin 3t \ -6 \sin 3t \end{pmatrix} + \begin{pmatrix} -\sin 3t \ 4 \sin 3t \ 3 \sin 3t \end{pmatrix}\]to get\[\begin{pmatrix} 3 \cos 3t \ 0 \ -3 \sin 3t \end{pmatrix}.\]

• Comparing this result to the computed derivative \( \mathbf{X}_{p}^{\prime}(t) \) confirms consistency.
• If both sides match, then the original vector \( \mathbf{X}_{p}(t) \) is indeed a true solution.

Consistency is the final step in validating that the particular solution satisfies the nonhomogeneous equation.