(Solved): 1 point) Consider the initial value problem \[ m y^{\prime \prime}+c y^{\prime}+k y=F(t), \quad y(0 ...

1 point) Consider the initial value problem \[ m y^{\prime \prime}+c y^{\prime}+k y=F(t), \quad y(0)=0, \quad y^{\prime}(0)=0 \] modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force \( F(t) \), where the unit of force is the Newton \( (\mathrm{N} \) ). Assume that \( m=2 \) kilograms, \( c=8 \) kilograms per second, \( k=80 \) Newtons per meter, and the applied force in Newtons is \[ F(t)=\left\{\begin{array}{ll} 40 & \text { if } 0 \leq t \leq \pi / 2 \\ 0 & \text { if } t>\pi / 2 \end{array}\right. \] a. Solve the initial value problem, using that the displacement \( y(t) \) and velocity \( y^{\prime}(t) \) remain continuous when the applied force is discontinuous. For \( 0 \leq t \leq \pi / 2, y(t)= \) help (formulas) For \( t>\pi / 2, y(t) \) Ip (formulas) b. Determine the long-term behavior of the system. Is \( \lim _{t \rightarrow \infty} y(t)=0 \) ? If it is, enter zero. If not, enter a function that approximates \( y(t) \) for very large positive values of \( t \). For very large positive values of \( t, y(t) \approx \) (formulas)