/*! 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} Problem 47 Determine whether the statement ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 47

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. The function \(f(x)=2 e^{x}-\frac{1}{2}(\cos x+\sin x)\) is a solution of the differential equation \(y^{\prime}-y=\sin x\).

Short Answer

Expert verified
The statement is False. After substituting \(f(x) = 2e^x - \frac{1}{2}(\cos x + \sin x)\) and its first derivative \(f'(x) = 2e^x - \frac{1}{2}\sin x + \frac{1}{2}\cos x\) into the differential equation \(y' - y = \sin x\), the resulting equation is \(\cos x = \sin x\), which is not equivalent to the original equation. Therefore, the function is not a solution to the given differential equation.

Step by step solution

01

Find the first derivative of f(x)

We need to find the first derivative of the function: $$ f(x) = 2e^x - \frac{1}{2}(\cos x+\sin x) $$ Deriving both terms separately, 1. Derivative of \(2e^x\): Using the chain rule, we have $$ \frac{d}{dx}(2e^x) = 2 \cdot e^x $$ 2. Derivative of \(\frac{1}{2}(\cos x + \sin x)\): Using the chain rule, we have $$ \frac{d}{dx}\left(\frac{1}{2}(\cos x + \sin x)\right) = -\frac{1}{2}\sin x+\frac{1}{2}\cos x $$ Adding these results together, we get the first derivative of \(f(x)\): $$ f'(x) = 2e^x - \frac{1}{2}\sin x + \frac{1}{2}\cos x $$
02

Plug f(x) and f'(x) into the differential equation

Now, we plug in \(f(x)\) and \(f'(x)\) into the given differential equation \(y' - y = \sin x\): $$ (2e^x - \frac{1}{2}\sin x + \frac{1}{2}\cos x) - (2e^x - \frac{1}{2}(\cos x + \sin x)) = \sin x $$
03

Simplify the equation

We now simplify the equation and see if it holds true: $$ 2e^x - \frac{1}{2}\sin x + \frac{1}{2}\cos x - 2e^x + \frac{1}{2}\sin x + \frac{1}{2}\cos x = \sin x $$ The terms \(2e^x\) and \(-2e^x\) cancel out, and we are left with: $$ \cos x = \sin x $$
04

Conclusion

Since the differential equation simplifies to \(\cos x = \sin x\), which is not equivalent to the original equation \(y'-y=\sin x\), the statement is False. The function \(f(x) = 2e^x - \frac{1}{2}(\cos x + \sin x)\) is not a solution to the differential equation \(y' - y = \sin x\).

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

The customer service department of Universal Instruments, manufacturer of the Galaxy home computer, conducted a survey among customers who had returned their purchase registration cards. Purchasers of its deluxe model home computer were asked to report the length of time \((t)\) in days before service was required. a. Describe a sample space corresponding to this survey. b. Describe the event \(E\) that a home computer required service before a period of 90 days had elapsed. c. Describe the event \(F\) that a home computer did not require service before a period of 1 yr had elapsed.

Let \(S\) be a sample space for an experiment. Show that if \(E\) is any event of an experiment, then \(E\) and \(E^{c}\) are mutually exclusive.

The manager of a local bank observes how long it takes a customer to complete his transactions at the automatic bank teller. a. Describe an appropriate sample space for this experiment. b. Describe the event that it takes a customer between 2 and 3 min to complete his transactions at the automatic bank teller.

In a survey of 200 employees of a company regarding their \(401(\mathrm{k})\) investments, the following data were obtained: 141 had investments in stock funds. 91 had investments in bond funds. 60 had investments in money market funds. 47 had investments in stock funds and bond funds. 36 had investments in stock funds and money market funds. 36 had investments in bond funds and money market funds. 22 had investments in stock funds, bond funds, and money market funds. What is the probability that an employee of the company chosen at random a. Had investments in exactly two kinds of investment funds? b. Had investments in exactly one kind of investment fund? c. Had no investment in any of the three types of funds?

A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. The sum of the numbers is an even number.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.