Second Translation Theorem (of the Laplace transform): L{uc(t)f(t - c)} = e-cs"L{f(t)} = e-cs F(s) L{uc(t)g(t)} = e-cs"L{g(t + c)} Examples L{uc(t)f(t - c)} = e-cs"L{f(t)} = e-cs F(s) This is typically used in reverse. Start with e-cs F(s) and follow the formula from there. Find the inverse Laplace transform of: e-4s !!!!!!!!!!!!" e-4s !!!!!!!!!!!!" (after partial fractions this becomes) e-4s (!!!! – !!!!) e-4s (4e2t – e-5t) u4(t) (4e2(t – 4) – e-5(t – 4)) u4(t) (4e2t - 8 – e-5t + 20) ""L {uc(t)g(t)} = e-cs"L{g(t + c)} This is typically used in the forward direction. Start with L{uc(t)g(t)} and follow the formula from there. Find the Laplace transform of: L{u4(t)t2e5t} L {u4(t)t2e5t} = e-4s"L{(t + 4)2e5(t + 4)} e-4s"L{(t2 + 8t +16)e5t+20} e-4s"e20 L {(t2 + 8t +16)e5t} e-4s"e20 (L {t2e5t}" "+"L {8te5t}" "+"L {16e5t}) e-4s"e20 (!(!!!)!"+ !(!!!)! +
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