BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Solutions

Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Solve the differential equation.

6.  d 2 y d x 2 + d y d x 2 y = x 2

To determine

To solve: The differential equation of d2ydx2+dydx2y=x2 .

Explanation

Given data:

The differential equation is,

d2ydx2+dydx2y=x2

y+y2y=x2 (1)

Formula used:

Write the expression for differential equation.

ay+by+cy=0 (2)

Write the expression for auxiliary equation.

ar2+br+c=0 (3)

Write the expression for general solution of ay+by+cy=0 with two distinct real roots.

yc(x)=c1er1x+c2er2x (4)

Here,

r1 and r2 are the roots of auxiliary equation.

Write the expression for the particular solution yp(x) .

yp(x)=Ax2+Bx+C (5)

Write the expression to find the roots of quadratic equation.

r=b±b24ac2a (6)

Compare equations (1) and (2).

a=1b=1c=2

Substitute 1 for a, 1 for b, and –2 for c in equation (3),

(1)r2+(1)r+(2)=0r2+r2=0

Find the roots of auxiliary equation using equation (6).

Substitute 1 for a, 1 for b, and –2 for c in equation (6),

r=(1)±(1)24(1)(2)2(1)=1±1+82=1±92=1±32

Consider the value of r1 and r2 as follows.

r1=1+32=22=1

r2=132=2

Substitute 1 for r1 and –2 for r2 in equation (4),

yc(x)=c1ex+c2e2x (7)

Differentiate equation (5) with respect to x,

yp(x)=ddx(Ax2+Bx+C)

yp(x)=2Ax+B (8)

Differentiate equation (8) with respect to x,

yp(x)=ddx(2Ax+B)

yp(x)=2A (9)

Substitute equations (5), (8) and (9) in equation (1),

2Ax+B+2A2(Ax2+Bx+C)=x2

2A+2Ax+B2Ax22Bx2C=x2 (10)

Substitute 0 for x in equation (10),

2A+2A(0)+B2A(0)

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Chapter 17 Solutions

Show all chapter solutions add
Sect-17.1 P-11ESect-17.1 P-12ESect-17.1 P-13ESect-17.1 P-14ESect-17.1 P-15ESect-17.1 P-16ESect-17.1 P-17ESect-17.1 P-18ESect-17.1 P-19ESect-17.1 P-20ESect-17.1 P-21ESect-17.1 P-22ESect-17.1 P-23ESect-17.1 P-24ESect-17.1 P-25ESect-17.1 P-26ESect-17.1 P-27ESect-17.1 P-28ESect-17.1 P-29ESect-17.1 P-30ESect-17.1 P-31ESect-17.1 P-32ESect-17.1 P-33ESect-17.1 P-34ESect-17.2 P-1ESect-17.2 P-2ESect-17.2 P-3ESect-17.2 P-4ESect-17.2 P-5ESect-17.2 P-6ESect-17.2 P-7ESect-17.2 P-8ESect-17.2 P-9ESect-17.2 P-10ESect-17.2 P-11ESect-17.2 P-12ESect-17.2 P-13ESect-17.2 P-14ESect-17.2 P-15ESect-17.2 P-16ESect-17.2 P-17ESect-17.2 P-18ESect-17.2 P-19ESect-17.2 P-20ESect-17.2 P-21ESect-17.2 P-22ESect-17.2 P-23ESect-17.2 P-24ESect-17.2 P-25ESect-17.2 P-26ESect-17.2 P-27ESect-17.2 P-28ESect-17.3 P-1ESect-17.3 P-2ESect-17.3 P-3ESect-17.3 P-4ESect-17.3 P-5ESect-17.3 P-6ESect-17.3 P-7ESect-17.3 P-8ESect-17.3 P-9ESect-17.3 P-10ESect-17.3 P-11ESect-17.3 P-12ESect-17.3 P-13ESect-17.3 P-14ESect-17.3 P-15ESect-17.3 P-16ESect-17.3 P-17ESect-17.3 P-18ESect-17.4 P-1ESect-17.4 P-2ESect-17.4 P-3ESect-17.4 P-4ESect-17.4 P-5ESect-17.4 P-6ESect-17.4 P-7ESect-17.4 P-8ESect-17.4 P-9ESect-17.4 P-10ESect-17.4 P-11ESect-17.4 P-12ECh-17 P-1RCCCh-17 P-2RCCCh-17 P-3RCCCh-17 P-4RCCCh-17 P-5RCCCh-17 P-1RQCh-17 P-2RQCh-17 P-3RQCh-17 P-4RQCh-17 P-1RECh-17 P-2RECh-17 P-3RECh-17 P-4RECh-17 P-5RECh-17 P-6RECh-17 P-7RECh-17 P-8RECh-17 P-9RECh-17 P-10RECh-17 P-11RECh-17 P-12RECh-17 P-13RECh-17 P-14RECh-17 P-15RECh-17 P-16RECh-17 P-17RECh-17 P-18RECh-17 P-19RECh-17 P-20RECh-17 P-21RE

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

U.S. GDP The gross domestic product (GDP) of the United States, in trillions of dollars, from 2011 through 2015...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Graph the curve and find the area that it encloses. 13. r = 2 + sin 4

Single Variable Calculus: Early Transcendentals

The appropriate option for the value of (123).

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False: converges.

Study Guide for Stewart's Multivariable Calculus, 8th