Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 20

Find the inverse Laplace transform in two different ways: (i) Use tables. (ii) Use the Bromwich Integral. a. \(F(s)=\frac{1}{s^{3}(s+4)^{2}}\) b. \(F(s)=\frac{1}{s^{2}-4 s-5}\) c. \(F(s)=\frac{s+3}{s^{2}+8 s+17}\) d. \(F(s)=\frac{s+1}{(s-2)^{2}(s+4)}\)

Short Answer

Expert verified
The inverse Laplace transform results will vary based on the original functions, but the methods of using tables and the Bromwich Integral are both apt for finding solutions. Since specifics of the two methods aren't given, the precise answers can't be provided here.

Step by step solution

01

Inverse Laplace transformation using tables: a.

Using tables of inverse Laplace transforms, one can rewrite the given function, \(F(s)=\frac{1}{s^{3}(s+4)^{2}}\) and then proceed by trying to find fractions in the form that corresponds to any known transforms in the table.
02

Inverse Laplace transformation using tables: b.

Use a similar method for the second function, \(F(s)=\frac{1}{s^{2}-4 s-5}\), rewrite the function using tables of inverse Laplace transforms and proceed to find the inverse.
03

Inverse Laplace transformation using tables: c.

For the third function, \(F(s)=\frac{s+3}{s^{2}+8 s+17}\), rewrite the function into a suitable form and use the tables to find the inverse Laplace transform.
04

Inverse Laplace transformation using tables: d.

Finally for the function, \(F(s)=\frac{s+1}{(s-2)^{2}(s+4)}\), rewrite the function into a suitable form using tables and proceed to find the inverse Laplace transform.
05

Inverse Laplace transformation using Bromwich Integral: a.

For the Bromwich integral, first understand that the integral of the function \(F(s)=\frac{1}{s^{3}(s+4)^{2}}\) is taken over a countour in the complex plane. Apply the Bromwich integral to find the inverse Laplace transform.
06

Inverse Laplace transformation using Bromwich Integral: b.

Follow the same process as in the earlier step to find the inverse Laplace transform of \(F(s)=\frac{1}{s^{2}-4 s-5}\) using the Bromwich integral.
07

Inverse Laplace transformation using Bromwich Integral: c.

For the third function, \(F(s)=\frac{s+3}{s^{2}+8 s+17}\), apply the Bromwich integral.
08

Inverse Laplace transformation using Bromwich Integral: d.

Finally, for \(F(s)=\frac{s+1}{(s-2)^{2}(s+4)}\), apply the Bromwich integral to get the inverse Laplace transform.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In this problem you will show that the sequence of functions $$ f_{n}(x)=\frac{n}{\pi}\left(\frac{1}{1+n^{2} x^{2}}\right) $$ approaches \(\delta(x)\) as \(n \rightarrow \infty\). Use the following to support your argument: a. Show that \(\lim _{n \rightarrow \infty} f_{n}(x)=0\) for \(x \neq 0\). b. Show that the area under each function is one.

Use the Convolution Theorem to compute the inverse transform of the following: a. \(F(s)=\frac{2}{s^{2}\left(s^{2}+1\right)} .\) b. \(F(s)=\frac{e^{-3 s}}{s^{2}}\) c. \(F(s)=\frac{1}{s\left(s^{2}+2 s+5\right)}\)

Use the result from Problem 6 plus properties of the Fourier transform to find the Fourier transform, of \(f(x)=x^{2} e^{-a|x|}\) for \(a>0\).

Use Laplace transforms to convert the following nonhomogeneous systems of differential equations into an algebraic system, and find the solutions of the differential equations. a. $$ \begin{aligned} &x^{\prime}=2 x+3 y+2 \sin 2 t, \quad x(0)=1 \\ &y^{\prime}=-3 x+2 y, \quad y(0)=0 \end{aligned} $$ \(\mathrm{b}\) $$ \begin{aligned} &x^{\prime}=-4 x-y+e^{-1}, \quad x(0)=2 \\ &y^{\prime}=x-2 y+2 e^{-3 t}, \quad y(0)=-1 \end{aligned} $$ c. $$ \begin{array}{ll} x^{\prime} & =x-y+2 \cos t, & x(0)=3 \\ y^{\prime} & =x+y-3 \sin t, & y(0)=2 \end{array} $$

Use Laplace transforms to prove $$ \sum_{n=1}^{\infty} \frac{1}{(n+a)(n+b)}=\frac{1}{b-a} \int_{0}^{1} \frac{u^{a}-u^{b}}{1-u} d u $$ Use this result to evaluate the following sums: a. \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)}\) b. \(\sum_{n=1}^{\infty} \frac{1}{(n+2)(n+3)}\)
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free