10. f(t) = egin{cases} 0, & 0 le t le 1 \ 1, & 1 < t le 5 \ 2, & t > 5 end{cases} (a) Rewrite f(t) using unit step functions. (b) Find mathcal{L}{f(t)}. (c) Use Laplace transforms to solve y' + y = f(t) subject to y(0) = 0.
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First, let's find the Laplace transform of f(t): \[\mathcal{L}\{f(t)\} = \mathcal{L}\{u(t - 1) - u(t - 5)\} = e^{-s} - e^{-5s}\] Now, let's find the Laplace transform of the differential equation: \[s^2Y(s) - sy(0) - y'(0) + Y(s) = e^{-s} - e^{-5s}\] Since y(0) = Show more…
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