Use the result of to show that where g(t) is periodic with
Chapter 7, Problem 44E(choose chapter or problem)
44. Use the result of Problem 43 to show that
\(\mathscr{L}^{-1}\left\{\frac{1}{\left(s^{2}+1\right)\left(1-e^{-\pi s}\right)}\right\}(t)=g(t)\)
Where \(g(t)\) is periodic with period \(2 \pi\) and \(g(t):=\begin{array}{ll}
\sin t, & 0 \leq t \leq \pi \\
0, & \pi \leq t \leq 2 \pi
\end{array}
\)
In Problems 45 and 46, use the method of Laplace transforms and the results of Problems 41 and 42 to solve the initial value problem.
\(y^{\prime \prime}+3 y^{\prime}+2 y=f(t) ; y(0)=0, y^{\prime}(0)=0\)
Where 𝒇(𝒕) is the periodic function defined in the stated problem.
Equation transcription:
Text transcription:
r{L}^{-1}{frac{1}{(s^{2}+1)(1-e^{-pi s})}\}(t)=g(t)
g(t)
2 pi
g(t):=begin{array}{ll}
sin t, & 0 leq t leq \pi \\
0, & \pi \leq t \leq 2 \pi
end{array}
y^{prime prime}+3 y^{prime}+2 y=f(t) ; y(0)=0, y^{prime}(0)=0
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