Chapter 7: Q7.3 - 23E (page 365)

Use Theorem 4on page362 to show how entry 32 follows from entry 31 in the Laplace transform table on the inside back cover of the text.

Short Answer

Expert verified

It is proved that, $L {t \sin b t + b t \cos b t} = \frac{2 b s^{2}}{{(s^{2} + b^{2})}^{2}}$ from the Laplace transform table.

Step by step solution

Define Laplace transform

When specific initial conditions are supplied, especially when the initial values are zero, the Laplace transform is a handy method of solving certain types of differential equations. Laplace transform $L$ of a function f(t) is defined as:

$L {f (t)} = \int_{0}^{\frac{<}{>}} e^{- s t} f (t) d t$

In words, we can describe this expression as the Laplace transform of f(t) equals function F of s, that is, $L {f (t)} = F (s)$ .

Show that L{tsinbt+btcosbt}=2bs2s2+b22

Consider the expression $L {t \sin b t + b t \cos b t}$

Let $f (t) = t \sin b t$

Then, $f' (t) = t \sin b t + b t \cos b t$

Find, $L {t \sin b t + b t \cos b t}$ using $L \{f'\} (s) = s L {f} (s) - f (0)$ and $L {t \sin b t} = \frac{2 b s}{{(s^{2} + b^{2})}^{2}}$ as:

$\begin{array}{rcl} L {t \sin b t + b t \cos b t} & = & s L {(t \sin b t)} - f (0) \\ = & s \frac{2 b s}{{(s^{2} + b^{2})}^{2}} - 0 \\ = & \frac{2 b s^{2}}{{(s^{2} + b^{2})}^{2}} \end{array}$

Hence, it is proved that $L {t \sin b t + b t \cos b t} = \frac{2 b s^{2}}{{(s^{2} + b^{2})}^{2}}$ .

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!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use Theorem 4on page362 to show how entry 32 follows from entry 31 in the Laplace transform table on the inside back cover of the text.

Short Answer

Step by step solution

Define Laplace transform

Show that L{tsinbt+btcosbt}=2bs2s2+b22

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Logic and Functions

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.