Chapter 8: Q16P (page 439)

Use L32 and L3 to obtain L11
$L {t s i n a t} = \frac{2 a p}{{(p^{2} + a^{2})}^{2}}$

Short Answer

Expert verified

Answer

The solution is $L {t \sin at} = \frac{2 ap}{{(p^{2} + a^{2})}^{2}}$ . So, given function is proved.

Step by step solution

Given information

The given function is $L {t \sin at} = \frac{2 ap}{{(p^{2} + a^{2})}^{2}}$

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)

Properties used to prove a given function

$L (32) : L {t^{n} g (t)} = {(- 1)}^{x} \frac{d^{4} G (p)}{d r^{n}} L (3) : L (\sin a t) = \frac{w}{p^{2} + a^{2}}$

Convert L32 and L3 into L11

Now, consider the Laplace $L \{x \sin a t\}$ .

role="math" localid="1654145096715" $L \{t^{π} g (t)\} = {(- 1)}^{n} \frac{d^{n} G (p)}{d p^{n}} L \{t^{1} \sin a t\} = {(- 1)}^{1} \frac{d}{d p} (\frac{a}{p^{2} + a^{2}})$

Differentiate with quotient rule, $\frac{d}{d x} (\frac{f}{z}) = \frac{T g - f g^{n}}{g^{2}}$

$L \{t^{1} \sin a t\} = (- \frac{d}{d p} (\frac{a}{p^{2} + a^{2}})) L \{t \sin a t\} = - [\frac{0 (p^{2} + a^{2}) - (p p) a}{{(p^{2} + a^{2})}^{2}}] L \{t \sin a t\} = - [\frac{- 2 a p}{{(p^{2} + a^{2})}^{2}}] L \{t \sin a t\} = \frac{2 a p}{{(p^{2} + a^{2})}^{2}}$

Hence $L \{t \sin a t\} = \frac{2 a p}{{(p^{2} + a^{2})}^{2}}$

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.

Use L32 and L3 to obtain L11
$L {t s i n a t} = \frac{2 a p}{{(p^{2} + a^{2})}^{2}}$

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Properties used to prove a given function

Convert L32 and L3 into L11

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Turning Points in Physics

Engineering Physics

Physical Quantities And Units

Wave Optics

Medical Physics

Electric Charge Field and Potential

Study anywhere. Anytime. Across all devices.

Use L32 and L3 to obtain L11L{tsinat}=2ap(p2+a2)2

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Properties used to prove a given function

Convert L32 and L3 into L11

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Turning Points in Physics

Engineering Physics

Physical Quantities And Units

Wave Optics

Medical Physics

Electric Charge Field and Potential

Study anywhere. Anytime. Across all devices.

Use L32 and L3 to obtain L11
$L {t s i n a t} = \frac{2 a p}{{(p^{2} + a^{2})}^{2}}$