Forced Oscillations & Resonance: x"+9x=10cos(2t),x(0)=x'(0)

[rooney]

express the solution of the given initial value problem as a sum of two oscillations:

x"+9x=10cos(2t) x(0)=x'(0)=0

If someone could take me through the steps of this it would be great. My book does not have a good example problem solved, my teacher hasn't lectured about this yet, and the solution manual that I have skips many steps. I really don't know where to begin with this one. Thanks.

 

[royhaas]

Re: Forced Oscillations and Resonance

Solve the homogeneous equation first, using the initial conditions.

 

[rooney]

Re: Forced Oscillations and Resonance

I found part of the general solution to be c_1cos(3t)+c_2sin(3t). The solution manual I'm referencing has the general solution equal to what i've found +2cos(2t).

Can someone instruct me on what to do to find this part? I know that I need to use the "special" function Acos(2t), but I can't figure out how to use.

 

[Deleted member 4993]

Re: Forced Oscillations and Resonance

I found part of the general solution to be c_1cos(3t)+c_2sin(3t). The solution manual I'm referencing has the general solution equal to what i've found +2cos(2t).

Can someone instruct me on what to do to find this part? I know that I need to use the "special" function Acos(2t), but I can't figure out how to use.

That is called particular solution of the ODE.

Do a google search with:

particular solution ODE

will find 1,870,000 sites ready to help you with example problems.

 

[rooney]

I don't understand the hesitation some have with assisting me. I've solved this problem, but I don't understand why I was instructed to do a google search for math help at a math help website.

 

[stapel]

I don't understand why I was instructed to do a google search for math help at a math help website.

Unfortunately, we really aren't able to teach lessons within this environment. You seemed to be requesting an explanation of the topic. Since this is something which probably comprised a fair number of pages in your textbook, and/or hours in the classroom, things we obviously cannot replicate here, you were directed toward a source of the requested lesson instruction.

I apologize for the confusion and for any offense our intended assistance has caused. My best wishes to you in your studies.

Eliz.

 

[rooney]

Re:

Unfortunately, we really aren't able to teach lessons within this environment. You seemed to be requesting an explanation of the topic. Since this is something which probably comprised a fair number of pages in your textbook, and/or hours in the classroom, things we obviously cannot replicate here, you were directed toward a source of the requested lesson instruction.

I guess I should have worded my question differently. I was hoping someone could solve this problem, and from the worked example I would be able to solve more on my own. Sorry if it seemed like I was asking for a lecture. I appreciate the explanation.

 

[Deleted member 4993]

Re: Re:

...I was hoping someone could solve this problem, and from the worked example I would be able to solve more on my own.

As you found out, from studying worked out examples in your text book (I hope), it is not that straight-forward. Particular solutions can take different forms - depending on nature of homogeneous solution and the forcing function. It is not possible to go into that detail in a page of this forum - where as a standard text book would take a chapter. So after studying those web-pages and your textbook - you are still in dark, you would need a face-to-face tutor.