   Chapter 17.2, Problem 12E

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+4y=ex (1)

Consider the auxiliary equation is,

r2+4=0 (2)

Roots of equation (2) are,

r=0±(0)24(1)(4)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=±4i2=±2i

Write the expression for the complementary solution for the complex roots,

yc(x)=eαx(c1cosβx+c2sinβx)

Substitute 0 for α and 2 for β ,

yc(x)=e0x(c1cos2x+c2sin2x)=c1cos2x+c2sin2x

If Right hand side (RHS) of a differential equation contains only exponential function.

Therefore, the trail solution yp(x) for this case can be expressed as follows

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