(Solved): 6. Consider the system depicted here. If \( x(t) \) denotes the displacement of a point to the lef ...

6. Consider the system depicted here. If \( x(t) \) denotes the displacement of a point to the left of the spring and \( y(t) \) the displacement of a point to the right of the spring, then \( x(t) \) and \( y(t) \) are related by the differential equation \[ b \dot{x}(t)+k x(t)=k y(t) . \] Let \( X(s) \) and \( Y(s) \) denote the Laplace transforms of \( x(t) \) and \( y(t) \), respectively. (a) Express \( X(s) \) as a function of \( x(0) \) and \( Y(s) \). 1 (b) Let \( b=k=1 \) and let \( x(0)=0 \). Suppose you were to stretch the spring steadily from the right so that \( y(t)=t u_{s}(t) \). Find \( x(t) \) for \( t \geq 0 \). [Recall that Laplace Transform of \( t u_{s}(t)=\frac{1}{s^{2}} \).]