Chapter 8: Q4P (page 439)

By differentiating the appropriate formula with respect to, verify L12.
$L (t \cos a t) = \frac{p^{2} - a^{2}}{{(p^{2} + a^{2})}^{2}}$
Answer

Short Answer

Expert verified

It is verified that

Step by step solution

Given information

The given function is $L (t \cos a t) = \frac{p^{2} - a^{2}}{{(p^{2} + a^{2})}^{2}}$

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 $f (t)$ .

Differentiate the given function

Let us consider the formula.

$L (\sin a t) = \int_{0}^{w} e^{- p t} \sin a t d t = \frac{a}{p^{2} + a^{2}}$

Differentiate both sides with respect to 'a ' as,

$\int_{0}^{=} e^{- p t} (t \cos a t) d t = \frac{d}{d u} [\frac{a}{p^{2} + a^{2}}]$

Use quotient rule, $\frac{d}{d a} (\frac{f}{g}) = \frac{f^{'} g - f g^{'}}{g^{2}}$

$\int_{0}^{π} e^{- p t} (t \cos a t) d t = \frac{d}{d a} (\frac{a}{p^{2} + a^{2}}) = \frac{\frac{d x}{d x} (p^{2} + a^{2}) - 2 a (a)}{{(p^{2} + a^{2})}^{2}} = \frac{p^{2} + a^{2} - 2 a^{2}}{{(p^{2} + a^{2})}^{2}} = \frac{p^{2} - a^{2}}{{(p^{2} + a^{2})}^{2}}$

Therefore,

Hence it is verified that $L (t \cos a t) = \frac{p^{2} - a^{2}}{{(p^{2} + a^{2})}^{2}}$

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.

Start your free trial

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

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

By differentiating the appropriate formula with respect to, verify L12.
$L (t \cos a t) = \frac{p^{2} - a^{2}}{{(p^{2} + a^{2})}^{2}}$
Answer

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Differentiate the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Oscillations

Waves Physics

Electrostatics

Measurements

Famous Physicists

Study anywhere. Anytime. Across all devices.

By differentiating the appropriate formula with respect to, verify L12.L(tcosat)=p2-a2p2+a22Answer

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Differentiate the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Oscillations

Waves Physics

Electrostatics

Measurements

Famous Physicists

Study anywhere. Anytime. Across all devices.

By differentiating the appropriate formula with respect to, verify L12.
$L (t \cos a t) = \frac{p^{2} - a^{2}}{{(p^{2} + a^{2})}^{2}}$
Answer