91Ó°ÊÓ

The forcing function \(f(t)\) is given as the sum of Dirac-delta functions, which consists of 20 sharp impulses at separate instances of time: \(1\pi, 2\pi,\dots, 20\pi\). These impulses will cause the undamped oscillator to gain energy instantaneously. The oscillator will likely experience significant oscillation during the impulse sequences. As there is no damping term, the oscillation of the system might continue indefinitely.

Now, let's find the actual solution to the given differential equation using Laplace Transforms. Applying the Laplace Transform to both sides of the given differential equation, we get: $$ s^{2}Y(s)-sy(0)-y^{\prime}(0)+Y(s) = F(s) $$ Since \(y(0)=0\) and \(y^{\prime}(0)=0\), the equation simplifies to: $$ (s^{2}+1)Y(s) = F(s) $$ The Laplace Transform of \(f(t)\) is: $$ F(s) = \sum_{k=1}^{20} \mathcal{L} \{\delta(t-k\pi)\} = \sum_{k=1}^{20} e^{-ks\pi} $$ Now, we can find the inverse Laplace Transform of the solution: $$ Y(s) = \dfrac{1}{s^{2}+1} \sum_{k=1}^{20} e^{-ks\pi} $$ Taking the inverse Laplace Transform, we get the solution for y(t): $$ y(t) = \mathcal{L}^{-1}\left\{ \dfrac{1}{s^{2}+1} \sum_{k=1}^{20} e^{-ks\pi} \right\} $$ Since the Laplace Transform is linear, we can split the sum into multiple inverse Laplace Transforms as: $$ y(t) = \sum_{k=1}^{20} \mathcal{L}^{-1}\left\{ \dfrac{e^{-ks\pi}}{s^{2}+1} \right\} = \sum_{k=1}^{20} \mathcal{U}_k(t) \sin{(t-k\pi)} $$ where \(\mathcal{U}_k(t)\) is the unit step function. We can now plot the solution y(t), as written above, to verify the behavior of the oscillator during and after the impulse sequences.

As seen in the solution found in step (b) above, the system oscillates for each impulse in the sum, where the oscillation starts at each \(k\pi\) with unit step functions. Since there is no damping term, once the impulses end, the system continues oscillating, but it won't receive any additional energy. Due to the linearity of the problem, the oscillations will either continue indefinitely or cease depending on the superposition of individual oscillations. Regardless, this behavior should align with our prediction in part (a) that the system might continue oscillating even after the sequence of impulses ends.