/*! 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} Q53E In Problems \(5\) and \(6\) comp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Q53E (page 526)

In Problems \(5\) and \(6\) compute \(y'\) and \(y''\) and then combine these derivatives with \(y\) as a linear second-order differential equation that is free of the symbols \({c_1}\) and \({c_2}\) and has the form \(F(y',y'',y''') = 0\). The symbols \({c_1}\) and \({c_2}\) represent constants.

\(y = {c_1}{e^x} + {c_2}x{e^x}\)

Short Answer

Expert verified

The answer is \(y'' - 2y' + y = 0\).

Step by step solution

01

Define second derivative of a function.

The derivative of the derivative of a function f is known as the second derivative, or second order derivative, in calculus.

So, the second derivative, or the rate of change of speed with respect to time, can be used to determine the variation in speed of the car (the second derivative of distance travelled with respect to the time).

02

Determine the second derivative of the function.

Let the first derivative of the given function be,

\(y' = {c_1}{e^x} + {c_2}x{e^x} + {c_2}{e^x}\)

Let the second derivative of the given function be,

\(y'' = {c_1}{e^x} + {c_2}x{e^x} + 2{c_2}{e^x}\)

Add \(y\)on both sides of the equation and factor out \(2\)from the equation.

\(\begin{array}{c}y'' + y = 2\left( {{c_1}{e^x} + {c_2}x{e^x}} \right) + 2{c_2}{e^x}\\ = 2\left( {{c_1}{e^x} + {c_2}x{e^x} + {c_2}{e^x}} \right)\\ = 2y'\end{array}\)

Hence, \(y'' - 2y' + y = 0\).

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

In Problems \(11 - 14\) verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution.

\(y'' + y = tanx; y = - (cosx) ln(secx + tanx)\).

In Problems 27鈥�30 use (12) of Section 1.1 to verify that the indicated function is a solution of the given differential equation. Assume an appropriate interval I of definition of each solution.

\(\frac{{dy}}{{dx}} + (sinx)y = x; y = {e^{cosx}}\int_0^x t {e^{ - cosx}}dt\)

In Problems \(15 - 18\) verify that the indicated function \(y = \phi (x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example \(6\), by considering \(\phi \) simply as a function and give its domain. Then by considering \(\phi \) as a solution of the differential equation, give at least one interval \(I\) of definition.

\(y' = 2x{y^2};y = 1/(4 - {x^2})\)

In Problems \(1\) and \(2\) Fill in the blank and then write this result as a linear first-order differential equation that is free of the symbol \({c_1}\) and has the form \(dy/dx = f(x,y)\). The symbol \({c_1}\) represents a constant.

In Problems \(19\) and \(20\) verify that the indicated expression is an implicit solution of the given first-order differential equation. Find atleast one explicit solutionin each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval \(I\) of definition of each solution \(\phi \).

\(2xydx + ({x^2} - y)dy = 0; - 2{x^2}y + {y^2} = 1\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics 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.