Chapter 7, Problem 117RE

True or False? Justify your answer with a proof or a counterexample.

117. If y and z are both solutions to $y^{″}+2y^{\prime }+y=0$ , then $y+z$ is also a solution.

To determine

To find:

The given statement true or false and to justify the answer.

Answer to Problem 117RE

The statement

$\text{"if y and z are both solutions to the }y\text{'}\text{'}+2y\text{'}+y=0\text{ then }y+z\text{ is also a solution"}$

Is true.

Explanation of Solution

Given:

The statement

$\text{"if y and z are both solutions to the }y\text{'}\text{'}+2y\text{'}+y=0\text{ then }y+z\text{ is also a solution"}$

Calculation:

From the given data

Consider the given differential equation $y\text{'}\text{'}+2y\text{'}+y=0$

$\text{ y and z are both solutions to the above differential equation}$

Then

$\begin{array}{l}y\text{'}\text{'}+2y\text{'}+y=0\\ \text{ z}\text{'}\text{'}+2z\text{'}+z=0\end{array}$

Let us assume $y+z$ is solution to $y\text{'}\text{'}+2y\text{'}+y=0$

Then

$\begin{array}{l}y\text{'}\text{'}+2y\text{'}+y=0\\ \Rightarrow \text{ (}y+z)\text{'}\text{'}+2(y+z)\text{'}+y+z\\ \Rightarrow y\text{'}\text{'}+z\text{'}\text{'}+2y\text{'}+2z\text{'}+y+z\\ \Rightarrow y\text{'}\text{'}+2y\text{'}+y+z\text{'}\text{'}+2z\text{'}+z\\ \Rightarrow 0\\ y\text{'}\text{'}+2y\text{'}+y=0\end{array}$

Therefore,

$y+z$ is solution to $y\text{'}\text{'}+2y\text{'}+y=0$

The given statement is true.