91Ó°ÊÓ

Find the Laplace transforms of the functions given in Problems 11 through \(22 .\) $$ f(t)=\sin \pi t \text { if } 2 \leqq t \leqq 3 ; f(t)=0 \text { if } t<2 \text { or if } t>3 $$

Use partial fractions to find the inverse Laplace transforms of the functions. \(F(s)=\frac{s^{3}}{(s-4)^{4}}\)

Use Laplace transforms to solve the initial value problems. \(x^{\prime \prime}-6 x^{\prime}+8 x=2 ; x(0)=x^{\prime}(0)=0\)

Suppose that \(f(t)\) is a periodic function of period \(2 a\) with \(f(t)=t\) if \(0 \leqq t

Use Laplace transforms to solve the initial value problems. \(x^{(4)}+8 x^{\prime \prime}+16 x=0 ; x(0)=x^{\prime}(0)=x^{\prime \prime}(0)=0\), \(x^{(3)}(0)=1\)

Show that the function \(f(t)=\sin \left(e^{t^{2}}\right)\) is of exponential order as \(t \rightarrow+\infty\) but that its derivative is not.

Use Laplace transforms to solve the initial value problems. \(x^{(4)}+2 x^{\prime \prime}+x=e^{2 t} ; x(0)=x^{\prime}(0)=x^{\prime \prime}(0)=x^{(3)}(0)=0\)

In Problems, apply the convolution theorem to derive the indicated solution \(x(t)\) of the given differential equation with initial conditions \(x(0)=x^{\prime}(0)=0 .\) $$ \begin{aligned} &x^{\prime \prime}+4 x^{\prime}+13 x=f(t) \\ &x(t)=\frac{1}{3} \int_{0}^{t} f(t-\tau) e^{-2 \tau} \sin 3 \tau d \tau \end{aligned} $$