solve the following system of linear non-homogeneous differential equations with eigenvalues/eigenvectors and variation of parameters
Transcribed Image Text:The image presents a system of differential equations in matrix form. It is expressed as follows:
\[
\frac{d\mathbf{z}}{d\mathbf{t}} = \begin{bmatrix} 0 & 2 \\ -1 & 3 \end{bmatrix} \mathbf{z} + \begin{bmatrix} 2 \\ e^{-3t} \end{bmatrix}
\]
where
\[
\frac{d\mathbf{z}}{d\mathbf{t}} = \begin{bmatrix} z_1' \\ z_2' \end{bmatrix}
\]
and
\[
\mathbf{z} = \begin{bmatrix} z_1 \\ z_2 \end{bmatrix}
\]
This system describes how the vector \(\mathbf{z}\), consisting of the components \(z_1\) and \(z_2\), evolves over time \(t\). The matrix on the right-hand side contains constant coefficients, determining how these components interact. The additional vector represents external input or forcing terms, with an exponential term \(e^{-3t}\) indicating a time-dependent influence.
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
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![The image presents a system of differential equations in matrix form. It is expressed as follows:
\[
\frac{d\mathbf{z}}{d\mathbf{t}} = \begin{bmatrix} 0 & 2 \\ -1 & 3 \end{bmatrix} \mathbf{z} + \begin{bmatrix} 2 \\ e^{-3t} \end{bmatrix}
\]
where
\[
\frac{d\mathbf{z}}{d\mathbf{t}} = \begin{bmatrix} z_1' \\ z_2' \end{bmatrix}
\]
and
\[
\mathbf{z} = \begin{bmatrix} z_1 \\ z_2 \end{bmatrix}
\]
This system describes how the vector \(\mathbf{z}\), consisting of the components \(z_1\) and \(z_2\), evolves over time \(t\). The matrix on the right-hand side contains constant coefficients, determining how these components interact. The additional vector represents external input or forcing terms, with an exponential term \(e^{-3t}\) indicating a time-dependent influence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d0ee2cb-cfe2-4eb9-a097-d38324436758%2F8316e925-53df-4e76-84ff-4e95610add4e%2Fiyj9w66_processed.jpeg&w=3840&q=75)
