Chapter 7: The Laplace Transform

Write each function in terms of unit step functions. Find the Laplace transform of the given function.

$f\left(t\right)=\left\{\begin{array}{ll}sint,& 0⩽t<2\pi \\ 0,& t⩾2\pi \end{array}\right.$

Solve the model for a driven spring/mass system with damping

$\mathbf{m}\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{x}}{{\mathbf{dt}}^{\mathbf{2}}}\mathbf{+}\mathbf{\beta }\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{+}\mathbf{kx}\mathbf{=}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{,}\mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{x}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

where the driving function f is as specied. Use a graphing utility to graph x(t) for the indicated values of t$\mathbf{m}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\mathbf{\beta }\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{k}\mathbf{=}\mathbf{5}\mathbf{,}\mathbf{f}$. is the meander function in Problem 53 with amplitude 10, and$\mathbf{a}\mathbf{=}\pi ,\mathbf{0}⩽\mathbf{t}⩽\mathbf{2}\pi$

Write each function in terms of unit step functions. Find the Laplace transform of the given function.

Solve the model for a driven spring/mass system with damping

$\mathbf{m}\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{x}}{{\mathbf{dt}}^{\mathbf{2}}}\mathbf{+}\mathbf{\beta }\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{+}\mathbf{kx}\mathbf{=}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{,}\mathbf{}\mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{x}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

where the driving function f is as specie. Use a graphing utility to graph x(t) for the indicated values of t.$\mathbf{m}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{\beta }\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{k}\mathbf{=}\mathbf{1}$ 1, f is the square wave in Problem 54 with amplitude 5, and $\mathbf{a}\mathbf{=}\pi ,\mathbf{0}⩽\mathbf{t}⩽\mathbf{4}\pi$

In Problems 55 – 62 write each function in terms of unit stepfunctions. Find the Laplace transform of the given function62.

Discuss how Theorem 7.4.1 can be used to find

${\mathbf{L}}^{\mathbf{-}\mathbf{1}}\left\{\mathbf{ln}\frac{\mathbf{s}\mathbf{-}\mathbf{3}}{\mathbf{x}\mathbf{+}\mathbf{1}}\right\}$

In Problems 63 – 70 use the Laplace transform to solve the given initial-value problem.

$y\text{'}+y=f\left(t\right),,y\left(0\right)=0wheref\left(t\right)=\left\{\begin{array}{ll}0,& 0⩽t<1\\ 5,& t⩾1\end{array}\right.$

In Section 6.4 we saw that$\mathbf{ty}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{=}\mathbf{0}$is Bessel’s equation of order$\mathit{v}\mathbf{=}\mathbf{0}$. In view of$\left(24\right)$of that section $\mathbf{ty}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{ty}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{}\mathit{i}\mathit{s} \mathbf{y}\mathbf{=}{\mathbf{J}}_{\mathbf{0}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$and Table 6.4.1 a solution of the initial-value problem. Use this result and the procedure outlined in the instructions to Problems$\mathbf{17}$and$\mathbf{18}$to show that.

$\mathbf{L}\left\{{\mathbf{J}}_{\mathbf{u}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\right\}\mathbf{=}\frac{\mathbf{1}}{\sqrt{{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{1}}}$

Use the Laplace transform to solve the given initial-value problem.

$y^{\text{'}}+y=f\left(t\right),y\left(0\right)=0,wheref\left(t\right)=\left\{\begin{array}{ll}1,& 0⩽t<1\\ -1,& t⩾1\end{array}\right.$

In Problems 63 – 70 use the Laplace transform to solve the given initial-value problem.

$y\text{'}+2y=f\left(t\right),y\left(0\right)=0,wheref\left(t\right)=f\left(t\right)=\{t,0\le t<10,t\ge 1$