Find the inverse laplace transforms for the following functions F(s)=frac{e^{-3s}}{s^2} F(s)=frac{2e^{-5s}}{s^2+4}

Find the inverse laplace transforms for the following functions F(s)=frac{e^{-3s}}{s^2} F(s)=frac{2e^{-5s}}{s^2+4}

Question
Laplace transform
asked 2021-02-13
Find the inverse laplace transforms for the following functions
\(F(s)=\frac{e^{-3s}}{s^2}\)
\(F(s)=\frac{2e^{-5s}}{s^2+4}\)

Answers (1)

2021-02-14
Step 1
Given
the given functions are
\(F(s)=\frac{e^{-3s}}{s^2}\)
\(F(s)=\frac{2e^{-5s}}{s^2+4}\)
Apply formula \(L^{-1}\left\{G(s)e^{-as}\right\}=g(t-a)u(t-a)\)
Step 2
\(F(s)=\frac{e^{-3s}}{s^2}\)
\(L^{-1}(\frac{1}{s^2})=t\)
Apply formula.
\(L^{-1}\left\{\left(\frac{1}{s^2}\right)e^{-3s}\right\}=(t−3)u(t−3)\)
Hence inverse Laplace transform of the function is \(F(t)=(t−3)u(t−3)\)
\(F(s)=\frac{2e^{-5s}}{(s^2+4)}\)
\(L^{-1}\left(\frac{2}{(s^2+4)}\right)=2L^{-1}\left(\frac{1}{(s^2+4)}\right)=2 \cdot \frac{\sin(2t)}{2}=\sin(2t)\)
Apply formula.
\(L^{-1}\left\{\left(\frac{2}{s^2+4}\right)e^{-5s}\right\}=\sin[2(t-5)]u(t-5)=\sin[2t-10]u(t-5)\)
Hence inverse Laplace transform of the function is \(F(t)=\sin[2t-10]u(t-5)\)
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