   Chapter 17.1, Problem 14E

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?14. 4   d 2 y d x 2 −   4 d y d x   +   y   =   0

To determine

To graph: The two basic solutions along with several other solutions of the differential equation for 4d2ydx24dydx+y=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 general solution of ay+by+cy=0 with two same roots (r) .

y=c1erx+c2xerx (3)

Here,

r is the root of auxiliary equation.

Consider the differential equation as follows.

4d2ydx24dydx+y=0 (4)

Compare equation (1) and (4).

a=4b=4c=1

Find the auxiliary equation.

Substitute 4 for a , 4 for b and 1 for c in equation (2),

(4)r2+(4)r+(1)=04r24r+1=0(2r1)2=02r1=0

Solve for r .

2r=1r=12

Substitute 12 for r in equation (3),

y=c1ex2+c2xex2 (5)

Consider c1 and c2 have some constant value.

Equation (5) consists two functions f(x) and g(x) as follows.

f(x)=ex2 (6)

g(x)=xex2 (7)

Here,

f(x) and g(x) are two basic solutions.

Consider the constant value for c1 and c2 as follows.

c1=2c2=3

Substitute 2 for c1 and 3 for c2 in equation (5),

y=2ex2+3xex2 (8)

Consider the constant value for c1 and c2 as follows.

c1=1c2=3

Substitute 1 for c1 and 3 for c2 in equation (5),

y=ex23xex2 (9)

Consider the constant value for c1 and c2 as follows

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