- #1
Mike86
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Homework Statement
Consider the initial value problem:
x'' + 2x' + 5x = δ(t - 1); with: x(0) = 0 and x'(0) = 0.
Using Laplace transforms, solve the initial value problem for x(t).
Homework Equations
L[x''] = (s^2)*L[x] - s*x(0) - x'(0)
L[x'] = s*L[x] - x(0)
L[δ(t - 1)] = e^(-s)
The Attempt at a Solution
Using the above known Laplace Transformations and the initial conditions I have gotten:
x'' + 2x' + 5x = δ(t - 1); with: x(0) = 0 and x'(0) = 0.
L[x''] + 2L[x'] + 5[x] = L[δ(t - 1)]
(s^2)*L[x] - s*x(0) - x'(0) + 2 (s*L[x] - x[0]) + 5 (L[x]) = e^(-s)
(s^2)*L[x] + 2s*L[x] + 5*L[x] = e^(-s)
(s^2 + 2s + 5)*L[x] = e^(-s)
L[x] = e^(-s) / (s^2 + 2s + 5)
From here I am not sure what Laplace Transformation to use to get the answer x. I can't really factorize (s^2 + 2s +5) because I would have to use the quadratic formula and I would get solutions with imaginary parts (from where I have no idea where to go as far as Laplace transformations are concerned).
I'm not sure if I made a mistake in the lead up (I can't see where) or there is a way to continue from here with the quadratic formula. Any advice with be immensely appreciated. Thanks!