\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \\ \dot{x}_4 \\ \dot{x}_5 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & -100 & -132 & -61 & -12 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} r(t) y = \begin{bmatrix} 100 & 8 & 72 & 18 & 2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{bmatrix}
Added by Walter H.
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Step 1
The Laplace transform of y(t) is given by Y(s) = 100/s^3 + 8/s^2 + 72/s + 18 + 2s. To find the partial fraction decomposition, we need to factor the denominator of Y(s) and express it as a sum of simpler fractions. The denominator of Y(s) is s^3 + 2s^2 + 72s + Show more…
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1. Solve the following equations using the Laplace transform. (a) y'' + y = 1, y(0) = 1, y'(0) = -1. (b) y''' - 2y' - 4y = 0, y(0) = 0, y'(0) = 1, y''(0) = 0 (Hint: s^2 + 2s + 2 = (s+1)^2 + 1)
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