91Ó°ÊÓ

Find the inverse Laplace transform \(f(t)\) of each function given in Problems 1 through \(10 .\) Then sketch the graph of \(f\). $$ F(s)=\frac{e^{-3 s}}{s^{2}} $$

Solve the initial value problems, and graph each solution function \(x(t)\). \(x^{\prime \prime}+4 x^{\prime}+4 x=1+\delta(t-2) ; x(0)=x^{\prime}(0)=0\)

Solve the initial value problems, and graph each solution function \(x(t)\). \(x^{\prime \prime}+2 x^{\prime}+x=t+\delta(t) ; x(0)=0, x^{\prime}(0)=1\)

Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph) in Problems 1 through \(10 .\) $$ f(t)=\cos t $$

Find the inverse Laplace transform \(f(t)\) of each function given in Problems 1 through \(10 .\) Then sketch the graph of \(f\). $$ F(s)=\frac{s e^{-s}}{s^{2}+\pi^{2}} $$

Solve the initial value problems, and graph each solution function \(x(t)\). \(x^{\prime \prime}+2 x^{\prime}+x=\delta(t)-\delta(t-2) ; x(0)=x^{\prime}(0)=2\)

Apply the translation theorem to find the inverse Laplace transforms of the functions. \(F(s)=\frac{s+2}{s^{2}+4 s+5}\)

Find the Laplace transforms of the functions given in Problems 11 through \(22 .\) $$ f(t)=1 \text { if } 1 \leqq t \leqq 4 ; f(t)=0 \text { if } t<1 \text { or if } t>4 $$

Find the Laplace transforms of the functions given in Problems 11 through \(22 .\) $$ f(t)=\sin t \text { if } 0 \leqq t \leqq 2 \pi ; f(t)=0 \text { if } t>2 \pi $$

Find the Laplace transforms of the functions given in Problems 11 through \(22 .\) $$ f(t)=\sin t \text { if } 0 \leqq t \leqq 3 \pi ; f(t)=0 \text { if } t>3 \pi $$