Q22P Obtain L(te-at聽cos聽bt)... [FREE SOLUTION]

91影视

Chapter 8: Q22P (page 439)

Obtain $L (t e^{- a t} c o s b t)$

Short Answer

Expert verified

Answer

The solution is $L \{t e^{- a t} \cos b t\} - \frac{{(p + a)}^{2} - b^{2}}{{({(蠁 + a)}^{2} + b^{2})}^{2}}$

Step by step solution

Given information

The given function is $f (t) = t e^{- a t} \cos b t$ and to prove is $L (t e^{a t} \cos b t)$

Definition of Laplace Transformation

A transformation of a function f(x) into the function g(t) that is useful especially in reducing the solution of an ordinary linear differential equation with constant coefficients to the solution of a polynomial equation.

The inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t)

Properties used for given function

$L (29) : L \{e^{- a t} g (t)\} = G (p + a) L (12) : L \{t \cos a t\} = \frac{p^{2} - a^{2}}{{(p^{2} + a^{2})}^{2}}$

Proof for given function

From L29

$L \{e^{- a i_{g}} g (t)\} = G (p + a)$

Since from L11

$L \{e^{- a i_{g}} g (t)\} = G (p + a)$

Consider $g (t) = t \cos b t$ and by use of (L29), L $\{e^{- a t} g (t)\} = G (p + a); L \{t e^{- w} \cos b t\}$ can be calculated by replace $p b y (p + a)$ in equation as,

$L \{t \cos b t\} = \frac{p^{2} - b^{2}}{{(p^{2} + b^{2})}^{2}} L \{t e^{- a t} \cos b t\} = \frac{{(p + a)}^{2} - b^{2}}{{({(p + a)}^{2} + b^{2})}^{2}}$

Hence, $L \{t_{e}^{- a t} \cos b t\} = \frac{{(p + a)}^{2} - b^{2}}{{({(p + a)}^{2})}^{2}}$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Obtain $L (t e^{- a t} c o s b t)$

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Properties used for given function

Proof for given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Conservation of Energy and Momentum

Further Mechanics and Thermal Physics

Electromagnetism

Linear Momentum

Kinematics Physics

Study anywhere. Anytime. Across all devices.

91影视

Obtain L(te-atcosbt)

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Properties used for given function

Proof for given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Conservation of Energy and Momentum

Further Mechanics and Thermal Physics

Electromagnetism

Linear Momentum

Kinematics Physics

Study anywhere. Anytime. Across all devices.

Obtain $L (t e^{- a t} c o s b t)$