Chapter 17.2, Problem 11E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Graph the particular solution and several other solutions. What characteristics do these solutions have in common?11. y" + 3y' + 2y = cos x    12. y" + 4y = e-x

To determine

To plot: The graph of the particular solution and several other solutions and found what characteristics of these solutions are common.

Explanation

Given data:

The differential equation is,

yâ€³+3yâ€²+2y=cosx (1)

Consider the auxiliary equation is,

r2+3r+2=0 (2)

Roots of equation (2) are,

r=âˆ’(3)Â±(3)2âˆ’4(1)(2)2(1)â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰{âˆµr=âˆ’bÂ±b2âˆ’4ac2aforâ€‰theâ€‰equationâ€‰ofar2+br+c=0â€‰â€‰}=âˆ’3Â±12=âˆ’1â€‰andâ€‰âˆ’2

Write the expression for the complementary solution for the real roots.

yc(x)=c1er1x+c2er2x

Substitute âˆ’1 for r1 and âˆ’2 for r2 ,

yc(x)=c1eâˆ’1x+c2eâˆ’2x

yc(x)=c1eâˆ’x+c2eâˆ’2x (3)

From equation (1), G(x)=cosx . Write the particular solution yp(x) for this case,

yp(x)=Acosx+Bsinx (4)

Differentiate equation (4) with respect to x,

yâ€²p(x)=ddx(Acosx+Bsinx)

yâ€²p(x)=âˆ’Asinx+Bcosx (5)

Differentiate equation (5) with respect to x,

yâ€³p(x)=ddx(âˆ’Asinx+Bcosx)

yâ€³p(x)=âˆ’Acosxâˆ’Bsinx (6)

Substitute equations (4), (5) and (6) in (1),

(âˆ’Acosxâˆ’Bsinx)+3(âˆ’Asinx+Bcosx)+2(Acosx+Bsinx)=cosx

(A+3B)cosx+(âˆ’3A+

### 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