Question

Consider an input function given by f(t)={3/2t−5​0≤t<4/t≥4​ a) Find the Laplace Transform of the input function. b) Obtain the response of the following system to such a forcing function given that x(0)=1, and x′(0)=0. x¨(t)+x(t)=f(t)

Answer 1

To find the Laplace Transform of the input function, split it into two parts: one for t less than 4 and one for t greater than or equal to 4. Obtain the response of the system by substituting the Laplace Transform into the differential equation.

Step-by-step explanation:

To find the Laplace Transform of the input function, we need to split it into two parts: one for t less than 4 and one for t greater than or equal to 4. For the first part, we have f(t) = 3/2t - 5. To find its Laplace Transform, we use the general formula for the Laplace Transform of a power function: L{t^n} = n!/(s^(n+1)). Applying this formula, we get L{3/2t - 5} = (3/2)*L{t} - 5/s^2. For the second part of the function, we have f(t) = 0. Its Laplace Transform is simply 0. Therefore, the overall Laplace Transform of the input function is (3/2)*L{t} - 5/s^2.

To obtain the response of the system to the input function, we substitute the Laplace Transform of the function into the differential equation x''(t) + x(t) = f(t). This gives us x''(t) + x(t) = (3/2)*L{t} - 5/s^2. Solving this differential equation will give us the response of the system.