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Q22E

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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 391
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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Short Answer

solve the given initial value problem using the method of Laplace transforms. Sketch the graph of the solution.

w''+w=u(t--2)--u(t--4);w(0)=1,w'(0)=0

On solving the given initial value problem using the method of Laplace transforms,the solution isw(t)=cost+[1-cos(t-2)]u(t-2)-[1-cos(t-4)]u(t-4) and the corresponding graph is

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Definition

The Laplace transform, is an integral transform that converts a function of a real variable usually t, the time domain to a function of a complex variable s.

Step 2: Taking Laplace Transform of initial value Problem 

w''+w=u(t-2)-u(t-4)

Where w(0)=1 and w'(0)=0

Lw''(s)+Lw(s)=L[u(t-2)-u(t-4)]s2Lw(s)-sLw(0)-w'(0)+Lw(s)=e-2ss-e-48ss2Lw(s)-s-0+Lw(s)=e-2ss-e-48s

s2+1Ly(s)-s=e-28s-e-18sLw(s)=ss2+1+e-2sss2+1-e4sss2+1

Step 3: By Partial fraction

1ss2+1=1s-ss2+1

Equation 1 becomes,

Lw(s)=ss2+1+e-2ss-se-2ss2+1-e-4ss+se-4ss2+1

Step 4: Taking inverse Laplace transform

w(t)=L-1ss2+1+L-1e-2ss-L-1se-2ss2+1-L-1e-4ss+L-1se-4ss2+1=cost+u(t-2)-cos(t-2)u(t-2)-u(t-4)+cos(t-4)u(t-4)=cost+[1-cos(t-2)]u(t-2)-[1-cos(t-4)]u(t-4)

Hence, w(t)=cost+[1-cos(t-2)]u(t-2)-[1-cos(t-4)]u(t-4)

, the graph is given below

Most popular questions for Math Textbooks

In Problems 1-14 , solve the given initial value problem using the method of Laplace transforms.

y''-y=t-2;y2=3, y'2=0

In Problems 1-14 , solve the given initial value problem using the method of Laplace transforms.

3.y''+6y'+9y=0; y0=-1, y'0=6

In Problems 3–10, determine the Laplace transform of the given function.

e2t-t3+t2-sin5t

In Problems 25 - 32, solve the given initial value problem using the method of Laplace transforms.

z''+3z'+2z=e-3tu(t-2)z(0)=2, z'(0)=-3

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

x'+y=1−u(t−2);x(0)=0,x+y'=0;y(0)=0
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