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Question 5 (20 Points): A simplistic model of a pair of conven-tional forces in combat yields the following system: \mathbf{x}^{\prime}=\mathbf{A x}+\mathbf{F} where \mathbf{x}^{\prime}=\left[\begin{array}{l}

x_{1}^{\prime}(t) \\

x_{2}^{\prime}(t)

\end{array}\right], \quad \mathbf{x}=\left[\begin{array}{l}

x_{1}(t) \\

x_{2}(t)

\end{array}\right], \quad \mathbf{A}=\left[\begin{array}{cc}

-a & -b \\

-c & -d

\end{array}\right], \quad \text { and } \mathbf{F}=\left[\begin{array}{l}

p \\

q

\end{array}\right] The variables 1₁(t) and x₂(t) represent the strengths of opposing forces at time t. The terms -ax₁ and -dx₂ represent the operational loss rates, and the terms -br2 and -cx₁ represent the combat loss rates for the troops ₁ and £2, respectively. The constants р and represent the respective rates of reinforcement. Let a = 1, b = 4,c = 3, d = 2, and p = q = 5. We will use the method of variation of-parameters to solve the system (2) and determine which forces will-win based on the given initial conditions. (a) Find the eigenvalues X₁ and λ₂ of the matrix A.

\text { (c) Write the fundamental matrix } \mathbf{X}(t)=\left[\begin{array}{ll}

e^{\lambda / t_{1}} \mathbf{u}_{1} & e^{\lambda_{2} t} \mathbf{u}_{2}

\end{array}\right] \text {. } \text { (d) Compute the inverse matrix } \mathbf{X}^{-1}(t) \text {. } \text { (e) Compute the product } \mathbf{X}^{-1}(t) \mathbf{F}(t) \text {. } \text { (f) Compute } X^{-1}(0) \text {. } For the initial condition x₁ = = x(0) =21/20 compute the solution of system 2 \mathbf{x}(t)=\mathbf{X}(t) \mathbf{X}^{-1}(0) \mathbf{x}_{0}+\mathbf{X}(t) \int_{0}^{t} \mathbf{X}^{-1}(s) \mathbf{F}(s) d s (h) Based on item (g), which forces will win. (Hint: Analyze the behavior of the solution (3) as t → +∞.)

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