Chapter 17.2, Problem 12E

### 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+4y=eâˆ’x (1)

Consider the auxiliary equation is,

r2+4=0 (2)

Roots of equation (2) are,

r=âˆ’0Â±(0)2âˆ’4(1)(4)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€‰}=Â±4i2=Â±2i

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

yc(x)=eÎ±x(c1cosÎ²x+c2sinÎ²x)

Substitute 0 for Î± and 2 for Î² ,

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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