   Chapter 17.1, Problem 15E

Chapter
Section
Textbook Problem

Graph the two basic solutions along with several other solutions of the differential equation. What features do the solutions have in common?15. d 2 y d x 2 + 2 d y d x   +   2 y   =   0

To determine

To graph: The two basic solutions along with several other solutions of the differential equation for d2ydx2+2dydx+2y=0 .

Explanation

Formula used:

Write the expression for differential equation.

ay+by+cy=0 (1)

Write the expression for auxiliary equation.

ar2+br+c=0 (2)

Write the expression for the complex roots.

r=α±iβ (3)

Write the expression for general solution of ay+by+cy=0 with complex roots.

y=eαx(c1cosβx+c2sinβx) (4)

Here,

α is the real part of the root, and

β is the imaginary part of the root.

Consider the differential equation as follows.

d2ydx2+2dydx+2y=0 (5)

Compare equation (1) and (5).

a=1b=2c=2

Find the auxiliary equation.

Substitute 1 for a , 2 for b and 2 for c in equation (2),

(1)r2+(2)r+(2)=0r2+2r+2=0

Solve for r .

r=(2)±(2)24(1)(2)2(1)=(2)±482=2±42=2±2i2

Simplify r as follows.

r=1±i (6)

Compare equations (3) and (6).

α=1β=1

Substitute 1 for α and 1 for β in equation (4),

y=e(1)x(c1cos(1)x+c2sin(1)x)

y=ex(c1cosx+c2sinx) (7)

Here,

c1 and c2 are constant value.

Consider the value of c1 and c2 as 1 .

Substitute 1 for c1 and 1 for c2 in equation (7),

y=ex((1)cosx+(1)sinx)=ex(cosx+sinx)

y=excosx+exsinx (8)

Equation (8) consists two functions f(x) and g(x) as follows

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