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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)=鈭�0g(t)路e-ptdt

Differentiate with respect to, we get

ddpG(p)=ddp鈭�0g^(t)-e-ptdt=鈭�0蔚g(t)路鈭�鈭�pe-mdt=鈭�00g(t)-tt-ptdt=-鈭�0-pe-p1(tg(t)dt

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

Hence Proved.

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Most popular questions from this chapter

Using (3.9), find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after (3.9), and Example 1.

(xlnx)y+y=Inx

For each of the following differential equations, separate variables and find a solution containing one arbitrary constant. Then find the value of the constant to give a particular solution satisfying the given boundary condition. Computer plot a slope field and some of the solution curves.

9 (1+y)y'=y, y=1Whenx=1

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By using Laplace transforms, solve the following differential equations subject to the given initial conditions.y''+y'-5y=e2t,鈥�y0=1,y0'=2

Use the results which you have obtained in Problems 21 and 22 to find the inverse transform of(p2+2p-1)I(p2+4p+5)2.
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