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Laplace Transform

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Solve the following DE using Laplace Transform. Be sure to show your steps and reference the Laplace Transform you will be using. {i.e. - e^at sin bt; b/((s-a)^2 + b^2) (s > a) }

(d^2y)/(dt^2) + 2(dy/dt) + 2y = x(t)

where

x(t) = { 1, 0 < = t < = 1
x(t) = { 2, t > = 2
x(t) = { 0, elsewhere

and

y(0) = 0; y'(0) = 1

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Hi,

x(t) in the given condition is equal to:
x(t) = u(t) + u(t-2)

We use Laplace transform to get the equation:
[s^2Y(s)-sy(0)-y'(0)] + 2[sY(s)-y(0)] + 2Y(s) = 1/s + e^(-2s)/s

Then we factor out Y(s) to get:
Y(s) [s^2 + 2s +2] - 1 = 1/s + e^(-2s)/s
Y(s) [s^2 + 2s +2] = 1/s + e^(-2s)/s + 1
Y(s) = (1+s)/[s(s^2 + 2s +2)] + e^(-2s)/[s(s^2 + 2s +2)]

Separating using partial fractions for the first ...

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