Chapter 8: Q26P (page 439)

Use L28 and L4 to find the inverse transform of $p e^{- p t} I (p^{2} + 1)$ .

Short Answer

Expert verified

Answer

The inverse Laplace transforms of is

$L^{- 1} [\frac{p^{- e z}}{p^{2} + 1}] = f (t) = \{\begin{cases} \cos (t - π), \\ 0, \end{cases} \begin{array}{r} t > π \\ t < π \end{array}\}$

Step by step solution

Given information

The given function is $p e^{- p 2} I (p^{2} + 1)$

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)

Inverse Transformation of the given function

From $(L 28)$

$L [f (t) = \{\begin{cases} g (t - a), \\ 0, \end{cases} \begin{array}{r} t > 0 \\ 0 \end{array}\}] = e^{- p w} G (p)$

From $(L 4)$

$L (\cos f) = \frac{p}{w^{2} + a^{2}}$

From above equation

$L (\cos t) = \frac{p}{p^{2} + 1}$

By use of $(L 28);$

$L [f (t) = \{\begin{array}{l} g (t - a), \\ 0, \end{array} \begin{array}{r} t > a > 0 \\ 0 \end{array}\}] = e^{- p u} G (p)$

And consider the

$L^{- 1} [e^{- p x} G (p)] = f (t) = \{\begin{array}{l} g (t - a), \\ 0, \end{array} \begin{array}{r} t > a > 0 \\ 0 \end{array}\} L^{- 1} [\frac{p x}{p^{2} + 1}] = f (t) = \{\begin{array}{l} g (t - π), \\ 0, \end{array} \begin{array}{r} t > π \\ t < π \end{array}\}$

Thus, the inverse Laplace transforms of $\frac{p \times r}{p^{2} + 1}$ is $L^{- 1} [\frac{p^{- e z}}{p^{2} + 1}] = f (t) = \{\begin{array}{l} \cos (t - π), \\ 0, \end{array} \begin{array}{r} t > π \\ t < π \end{array}\}$ .

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 L28 and L4 to find the inverse transform of $p e^{- p t} I (p^{2} + 1)$ .

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Inverse Transformation of the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Wave Optics

Thermodynamics

Conservation of Energy and Momentum

Measurements

Physics of Motion

Electricity

Study anywhere. Anytime. Across all devices.

Use L28 and L4 to find the inverse transform ofpe-ptI(p2+1).

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Inverse Transformation of the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Wave Optics

Thermodynamics

Conservation of Energy and Momentum

Measurements

Physics of Motion

Electricity

Study anywhere. Anytime. Across all devices.

Use L28 and L4 to find the inverse transform of $p e^{- p t} I (p^{2} + 1)$ .