(1 point) Take the Laplace transform of the following initial value and solve for Y(s) = mathcal{L}{y(t)}; y'' + 9y = egin{cases} sin(pi t), & 0 le t < 1 \ 0, & 1 le t end{cases} y(0) = 0, y'(0) = 0 Y(s) = Heaviside function. Now find the inverse transform to find y(t) = for u_c(t). Note: frac{pi}{(s^2 + pi^2)(s^2 + 9)} = frac{pi}{pi^2 - 9} left( frac{1}{s^2 + 9} - frac{1}{s^2 + pi^2} ight)
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Step 1:** Take the Laplace transform of the differential equation: \(s^2Y(s) - sY(0) - Y'(0) + 9Y(s) = \int_0^1 \sin(ct)dt\) \(s^2Y(s) - 9Y(s) = \frac{1}{s} - \frac{c}{s^2 + c^2}\) ** Show more…
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Take the Laplace transform of the following initial value and solve for Y(s) = L{y(t)}: y'' + 9y = { sin(̀́̂t), 0 <= t < 1; 0, 1 <= t. y(0) = 0, y'(0) = 0. Y(s) = Next, take the inverse transform of Y(s) to get y(t) = Use step(t-c) for uc(t). Note: ̀́̂ / ((s^2 + ̀́̂^2)(s^2 + 9)) = ̀́̂ / (̀́̂^2 - 9) * (1 / (s^2 + 9) - 1 / (s^2 + ̀́̂^2))
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