   Chapter 17.2, Problem 11E

Chapter
Section
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,

y+3y+2y=cosx (1)

Consider the auxiliary equation is,

r2+3r+2=0 (2)

Roots of equation (2) are,

r=(3)±(3)24(1)(2)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=3±12=1and2

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

yc(x)=c1er1x+c2er2x

Substitute 1 for r1 and 2 for r2 ,

yc(x)=c1e1x+c2e2x

yc(x)=c1ex+c2e2x (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,

yp(x)=ddx(Acosx+Bsinx)

yp(x)=Asinx+Bcosx (5)

Differentiate equation (5) with respect to x,

yp(x)=ddx(Asinx+Bcosx)

yp(x)=AcosxBsinx (6)

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

(AcosxBsinx)+3(Asinx+Bcosx)+2(Acosx+Bsinx)=cosx

(A+3B)cosx+(3A+

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