(1 point) In this problem you will solve the nonhomogeneous system $\vec{y'} = \begin{bmatrix} -3 & -5 \\ 5 & 3 \end{bmatrix} \vec{y} + \begin{bmatrix} 4 \\ -2 \end{bmatrix}$ A. Write a fundamental matrix for the associated homogeneous system Ψ = B. Compute the inverse Ψ⁻¹ = C. Multiply by $\vec{g}$ and integrate $\int \Psi^{-1} \vec{g} dt = $ (Do not include $c_1$ and $c_2$ in your answers). D. Give the solution to the system $\vec{y} = $ + (Do not include $c_1$ and $c_2$ in your answers)
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First, we find the eigenvalues and eigenvectors of A. The characteristic equation is given by: $det(A - \lambda I) = (-3 - \lambda)(3 - \lambda) - (-5)(5) = \lambda^2 + 25 = 0$ $\lambda = \pm 5i$ For $\lambda_1 = 5i$, $(A - 5iI)\vec{v}_1 = 0$: $\begin{bmatrix} -3 Show more…
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