/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} 16E Question: In Problems 15-24, sol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: 16E (page 350)

Question: In Problems 15-24, solve for Ys, the Laplace transform of the solution yt to the given initial value problem.

y''+6y=t2-1;y0=0,y'0=-1

Short Answer

Expert verified

The Initial value for y''+6y=t2-1isY=-s3-s2+2s3s2+6

Step by step solution

01

The Laplace Transform

  • The Laplace transform is a strong integral transform used in mathematics to convert a function from the time domain to the s-domain.
  • In some circumstances, the Laplace transform can be utilized to solve linear differential equations with given beginning conditions.
  • Fs=鈭�0鈭�f(t)e-stt'
02

Determine the Laplace transform

Define Lys=Ys

Using the properties listed below, take the Laplace transform of the equation.

Ly''s=s2Lys-sy0-y'0Ltns=n!sn+1

L1s=1sLy''+6Ly=Lt2-L1

Substitute the properties into the equation

s2Y-sy0-y'0+6Y=2!s3-1s

Substitute the initial conditions

y0=0andy'0=-1

s2Y+1+6Y=2s3-1s

Isolate the Y variable as:

Ys2+6=2s3-1s-1=-s3-s2+2s3

Y=-s3-s2+2s3s2+6

Therefore, the initial value fory''+6y=t2-1 isY=-s3-s2+2s3s2+6

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

Most popular questions from this chapter

Use Theorem 4on page362 to show how entry 32 follows from entry 31 in the Laplace transform table on the inside back cover of the text.

Find the transfer function, as defined in Problem 29, for the linear system governed by


y''(t)+5y'(t)+6y(t)=g(t),t>0.

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

x''+y=1;x(0)=1,x'(0)=1x+y''=-1;y(0)=1,y'(0)=-1

In Problems 3鈥�10, determine the Laplace transform of the given function.e3tsin4t

In Problems 1 and 2, use the definition of the Laplace transform to determineL{f}.

f(t)={3,0鈮�t鈮�26-t,2<t
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.