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 initial-value problem.

14. 9y" + y =3x + e-x, y(0) = 1, y'(0) = 2

To determine

To solve: The initial-value problem.

Explanation

Given data:

The initial-value problem is,

9y+y=3x+ex (1)

With y(0)=1 and y(0)=2 .

Consider the auxiliary equation.

9r2+1=0 (2)

Roots of equation (2) are,

r=0±(0)24(9)(1)2(9){r=b±b24ac2afortheequationofar2+br+c=0}=±i618=±13i

Write the expression for the complementary solution of two complex roots r=α±iβ ,

yc(x)=eαx(c1cosβx+c2sinβx) (3)

Substitute 0 for α and 13 for β in equation (3),

yc(x)=e0x(c1cos13x+c2sin13x)

yc(x)=c1cos13x+c2sin13x (4)

Re-arrange equation (1) by neglecting the term ex ,

9y+y=3x (5)

The Right hand side (RHS) of a differential equation contains a polynomial. Therefore, the trail solution yp(x) for this case can be expressed as follows.

yp1(x)=Ax+B (6)

Differentiate equation (6) with respect to x,

yp1(x)=ddx(Ax+B)

yp1(x)=A (7)

Differentiate equation (7) with respect to x,

yp2(x)=ddx(A)

yp1(x)=0 (8)

Substitute equations (6) and (8) in equation (1),

9(0)+Ax+B=3x

Ax+B=3x (9)

By equating coefficients of equation (9), consider the followings,

A=3

And

B=0

Substitute 3 for A and 0 for B in equation (6),

yp1(x)=3x+0

yp1(x)=3x (10)

Re-arrange equation (1) by neglecting the term 3x ,

9y+y=ex (11)

The Right hand side (RHS) of a differential equation contains only an exponential function. Therefore, the trail solution yp(x) for this case can be expressed as follows

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

Intervals Graph the set. 65. (, 4) (4, )

Precalculus: Mathematics for Calculus (Standalone Book)

In Exercises 1520, simplify the expression. 20. 81x5y1039xy23

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

Find the volume of each rectangular solid:

Elementary Technical Mathematics