Q19P Prove the general formula L29. ... [FREE SOLUTION]

91影视

Chapter 8: Q19P (page 429)

Prove the general formula L29.

Short Answer

Expert verified

Answer

The function $L (e^{- a t} g (f)) = G (p + a)$ is proved

Step by step solution

Given information

The given function which has to prove is $L (f) = {鈭�}_{0} f (t) e^{- p t} d t = F (p)$ and function which has to prove is $L (e^{- a t} g (f)) = G (p + a)$

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)

Proof for the given function

It can be calculated as,

$L (f) = {鈭�}_{0}^{鈭�} f (t) e^{- p t} d t L (e^{- a t} g (t)) = {鈭�}_{0}^{p} e^{- a t} e^{- p t} g (t) d t L (e^{- a t} g (t)) = {鈭�}_{0}^{p} e^{- a t} e^{- w - 渭 t} g (t) d t L (e^{- a t} g (t)) = {鈭�}_{0}^{鈭�} e^{- (p + u) 鈻�} g (t) d t$

Again, with use of

${鈭�}_{0}^{"} f (t) e^{- p} d t = F (p)$

$L (e^{- 2} (t))$ Can be calculated as,

$L (e^{- a t} g (t)) = {鈭�}_{0} e^{- (p + a)} g (t) d t = G (p + a)$

Hence, $L (e^{- a t} g (t)) = G (p + a)$ .

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

Prove the general formula L29.

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Proof for the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Particle Model of Matter

Circular Motion and Gravitation

Geometrical and Physical Optics

Torque and Rotational Motion

Thermodynamics

Modern Physics

Study anywhere. Anytime. Across all devices.