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(a) Find the general solution of the motion of a mass attached to the ceiling by a spring in presence of friction, i.e. solve the ODE \[ m \ddot{y}=m g-k(y-l)-\gamma \dot{y} . \] with \( m=1, k=3, \gamma=2, g=10, l=5 \), where \( y \) indicates the distance of the mass from the ceiling. \( [8] \) (b) What is the limit \( \lim _{t \rightarrow \infty} y(t) \) for the motion of the mass described in (a)? Describe in words the asymptotic dynamical behaviour of the mass for \( t \rightarrow \infty \). \( [4] \) (c) Determine whether the differential equation \[ \frac{1}{2} y^{2}+y \cos (x)+\left(y x+\sin (x)-e^{y}\right) y^{\prime}=0 \] is exact. If it is exact, find its general solution in explicit form. \( [14] \)

c) 12y2+ycos?(x)+(yx+sin?(x)?ey)y?=0 12y2+ycos?(x)+(yx+sin?(x)?ey)dydx=0 ?(12y2+ycos?(x))dx+(yx

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