Damped harmonic oscillator Diff. Eq. question

[Dusty912]

Homework Statement

consider any damped harmonic oscillator equation

m(d²t/dt² +bdy/dt +ky=0

a. show that a constant multiple of any solution is another solution
b. illustrate this fact using the equation
(d²t/dt² +3dy/dt +2y=0

c. how many solutions to the equation do you get uf you use this observatiob along with duess-and-test method described in this section?

Homework Equations

guess-and-test method. subbing e^st for y, and then solving for s value to obtain two solutions

The Attempt at a Solution

now I know for (a) I am supposed to sub in ky(t) into the damped harmonic oscillator. but I am stuck from there. I am not really sure what kind of answer this question is looking for. Any help would be appreciated.

 

[Buzz Bloom]

show that a constant multiple of any solution is another solution

H i Dusty:

If you assume y(t) is a solution, what do you think "a constant multiple of any solution is another solution" means?

Hope this helps.

Regards,
Buzz

 

[Dusty912]

well I believe it means that if y(t) is a solution then ky(t) is also a solution. Where k is an arbitrary constant.

 

[Dusty912]

Im just not sure what the question is seeking as far as written work

 

[Buzz Bloom]

Hi Dusty:

I would avoid using k as the "constant" used in the "constant multiple", since k appears in the equation. For example try cy(t) instead.
How would you show that cy(t) satisfies the DE?

Regards,
Buzz

 

[Dusty912]

Well by plugging it in. Correct?

 

[Buzz Bloom]

Hi Dusty:

What do you get when you plug it in?

Regards,
Buzz

 

[Dusty912]

cm d(y(t))/dt² +bc d(y(t))/dt +kcy(t)=0

 

[Buzz Bloom]

Hi Dusty:

Right. What do you get if you divide this equation by c? What does that tell you?

Regards,
Buzz