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An unforced linear time invariant (LTI) system is represented by [ẋ1; ẋ2] = [−1 0; 0 −2] [x1; x2]. If the initial conditions are x1(0) = 1 and x2(0) = −1, the solution of the state equation is

1. x1(t) = −1, x2(t) = 2
2. x1(t) = −e^(−t), x2(t) = 2e^(−t)
3. x1(t) = e^(−t), x2(t) = −e^(−2t)
4. x1(t) = −e^(−t), x2(t) = −2e^(−t)

Correct answer: x1(t) = e^(−t), x2(t) = −e^(−2t)

Solution

The correct option describes the solution to the state equation derived from the system's dynamics, where each state variable evolves independently according to its own exponential decay governed by the eigenvalues of the system matrix. The initial conditions lead to the specific forms of the solutions, confirming that x1(t) and x2(t) behave as indicated.

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