   Chapter 17, Problem 1RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If y1, and y2 are solutions of y" + y = 0, then y1 + y2 is also a solution of the equation.

To determine

Whether the statement “If y1 and y2 are solutions of y+y=0 , then y1+y2 is also a solution of the equation” is true or false.

Explanation

Given data:

y+y=0

Formula used:

Write the expression for solution of second order linear homogeneous differential equation.

y(x)=c1y1(x)+c2y2(x) (1)

Here,

c1 and c2 are constants,

y1(x) and y2(x) are solution of linear homogeneous equation.

Consider the second order linear homogeneous differential equation as follows.

y+y=0 (2)

Consider the general expression for second order linear homogeneous differential equation as follows.

P(x)y+Q(x)y+R(x)y=0 (3)

Compare equations (2) and (3).

P(x)=1Q(x)=0R(x)=1

The linear homogeneous differential equation consists of two solutions y and y represents as follows.

y=(c1y1+c2y2)y=(c1y1+c2y2)y=c1y1+c2y2

Modify equation (2) as follows.

y1+y1=0

Modify equation (2) as follows.

y2+y2=0

Consider right hand side (R.H.S) of equation (3)

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