Here’s a **structured breakdown** of the problem and step-by-step solution approach: --- # Laplace Transform Differential Equation Problem ## **Given:** A differential equation: \[ \frac{d^2x}{dt^2} + 6\frac{dx}{dt} + 8x = 3u_1(t) + u_2(t) \] **Initial conditions:** \(x() = 2\) \(\frac{dx}{dt}() = 1\) --- ## **Instructions:** ### **a. Take Laplace Transform** Apply the Laplace transform to both sides (using linearity and Laplace properties): Let \( X(s) = \mathcal{L}\{ x(t) \} \) \[ \mathcal{L}\left\{ \frac{d^2x}{dt^2} \right\} = s^2 X(s) - s x() - x'() \] \[ \mathcal{L}\left\{ \frac{dx}{dt} \right\} = s X(s) - x() \] \[ \mathcal{L}\{ x(t) \} = X(s) \] \[ \mathcal{L}\{ u_1(t) \} = \frac{1}{s} \] \[ \mathcal{L}\{ u_2(t) \} = \frac{1}{s} \] **Substitute values:** \[ s^2 X(s) - s \cdot 2 - 1 + 6 (s X(s) - 2) + 8 X(s) = 3 \frac{1}{s} + \frac{1}{s} \] **Expand and collect terms:** \[ s^2 X(s) - 2s - 1 + 6s X(s) - 12 + 8 X(s) = \frac{4}{s} \] Group \(X(s)\) terms: \[ (s^2 + 6s + 8) X(s) - 2s - 1 - 12 = \frac{4}{s} \] \[ (s^2 + 6s + 8) X(s) - 2s - 13 = \frac{4}{s} \] \[ (s^2 + 6s + 8) X(s) = \frac{4}{s} + 2s + 13 \] \[ X(s) = \frac{\frac{4}{s} + 2s + 13}{s^2 + 6s + 8} \] Or, combine: \[ X(s) = \frac{4 + (2s + 13)s}{s(s^2 + 6s + 8)} \] \[ X(s) = \frac{2s^2 + 13s + 4}{s(s^2 + 6s + 8)} \] **Factored denominator:** \( s(s+4)(s+2) \) So, \[ X(s) = \frac{2s^2 + 13s + 4}{s(s+2)(s+4)} \] --- ### **b. Predict the Final Value (Steady-State Output)** Use the **Final Value Theorem**: \[ \lim_{t \to \infty} x(t) = \lim_{s \to } s X(s) \] \[ \lim_{s \to } s \cdot \frac{2s^2 + 13s + 4}{s(s+2)(s+4)} = \lim_{s \to } \frac{2s^2 + 13s + 4}{(s+2)(s+4)} \] Substitute \(s = \): \[ \frac{4}{8} = .5 \] **Prediction:** The final value of \(x(t)\) as \(t \to \infty\) is **.5**. --- ### **c. Inverse Laplace Transform (MATLAB)** Use MATLAB to find the inverse Laplace: ```matlab syms s t X = (2*s^2 + 13*s + 4) / (s*(s+2)*(s+4)); x_t = ilaplace(X, s, t) ``` **Show MATLAB output and screenshot as required by your assignment.** --- ## **Summary Table** | Step | Task | Formula/Command | |------|-------------------------------------------|-------------------------------------------------------------| | a | Laplace Transform | \( X(s) = \frac{2s^2 + 13s + 4}{s(s+2)(s+4)} \) | | b | Final Value (Steady-State) | \( \lim_{s \to } s X(s) = .5 \) | | c | Inverse Laplace (in MATLAB) | `ilaplace(X, s, t)` | --- **Be sure to include a screenshot of your MATLAB workspace with your submission.**