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Chapter 8: Q15P (page 439)

Prove L32 forn=1. Hint: Differentiate equation (8.1) with respect top.

Short Answer

Expert verified

L32 for n=1 is=-d4p[G(p)]}

Step by step solution

01

Given information

The given function isL32

02

Definition of Laplace Transformation

A transformation of a function fxinto the function gtthat 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 functionFsis the piecewise-continuous and exponentially-restricted real functionft.

03

Differentiate the given function

Forn=1

Prove that,L(t-g(t))=-ddp[G(p)]

WhereG(p)=L(g(t))

We know thatG(p)=L(g(t))

G(p)=∫0g(t)·e-ptdt

Differentiate with respect to, we get

ddpG(p)=ddp∫0g^(t)-e-ptdt=∫0εg(t)·∂∂pe-mdt=∫00g(t)-tt-ptdt=-∫0-pe-p1(tg(t)dt

=-L[tg(t)]=-d4p[G(p)]

Hence Proved.

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Most popular questions from this chapter

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

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