Ok, need a little help in a couple of areas (2nd order w/forcing terms)

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Discussion Overview

The discussion revolves around solving second-order differential equations with forcing terms, specifically using the method of undetermined coefficients and exploring concepts related to resonance in oscillatory systems. Participants seek guidance on specific equations and methods, addressing both theoretical and practical aspects of the problems presented.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant describes a differential equation with a forcing term and attempts to find a particular solution using the method of undetermined coefficients, expressing difficulty in solving for certain coefficients.
  • Another participant suggests that resonance occurs when the forcing frequency matches the natural frequency of the system, indicating a potential approach to the first problem.
  • There is a discussion about the conditions under which oscillations increase without bound, with references to natural frequency and resonance.
  • Some participants propose using suitable trigonometric trial functions for finding particular solutions to the differential equations.
  • One participant expresses uncertainty about the relationship between the natural frequency and the forcing frequency, questioning if their understanding is correct.
  • Another participant shares their solution to a related problem using the annihilator method and seeks confirmation on the correctness of their answer.

Areas of Agreement / Disagreement

Participants generally agree on the concept of resonance and its implications for oscillatory systems. However, there is no consensus on the specific methods or solutions for the differential equations discussed, as participants express differing approaches and results.

Contextual Notes

Some discussions involve unresolved mathematical steps and varying methods for solving the equations, indicating potential limitations in the approaches taken by participants.

Who May Find This Useful

Students and practitioners in mathematics and engineering, particularly those dealing with differential equations and oscillatory systems, may find this discussion relevant.

amb123
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1) For one question, the forcing term is 8cos2x - 4sinx. I am trying to solve by the method of undetermined coefficients. The solution to the homogeneous equation is c1cosx + c2sinx so for the particular solution I was using :
Acos2x + Bsin2x + Cxcosx + Dxsinx, where I added the x's into the last two terms because without those they would be solutions of the homogenous equation. Where am I going wrong here? I end up able to solve for A and B, but not for C and D. I have tried using convolutions but the constants become messy, and there are many many terms to deal with.

2) Another question is an eq of motion w/o damping with a forcing term of sinwt (w=omega). the question is "for what value of omega do oscillations increase unbounded?". I am not sure exactly what I am looking for here. My book mentions briefly the consequences of something being underdamped, and the frequency matching and making bridges collapse, etc, but nothing like this question. Where do I begin to look for the guidance to solve this type of question?

Any/all guidance is appreciated.
thx.
-A
 
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it would help if you post the ode+(boundary conditios) as your description is rather vague.
 
Ok, sorry about that.

First question is x'' + 16x = sinwt which describes a mass-spring system under s sinusoidal driving force. the question is for what value of w do the oscillations of this system increase w/o bound. I just don't know exactly what I'm supposed to do, any hints on where to begin or what I'm looking for here?

another question is y'' + y = 8cos2x - 4sinx where y(pi/2) = -1 and y'(pi/2) = 0
I have tried this all different ways and keep having problems. I was just shown a new method earlier today and I'm going to try figure that one out this weekend, i'd be interested to find out if there is a method by which this is best solved and it works out well.

I'm also getting a nasty answer for y'' - 2y - 8y = 2e^-2x - e^-x where y(0)=y'(0) = 0. I get a really nasty answer using convolutions.

I'm most concerned with the first problem, though, because I can't figure out what I'm really trying to do with it.

thx for any help.
-A
 
"First question is x'' + 16x = sinwt which describes a mass-spring system under s sinusoidal driving force. the question is for what value of w do the oscillations of this system increase w/o bound. I just don't know exactly what I'm supposed to do, any hints on where to begin or what I'm looking for here?"
Hint:
When the forcing term hits upon the natural frequency of the system, you'll get resonance.

"another question is y'' + y = 8cos2x - 4sinx where y(pi/2) = -1 and y'(pi/2) = 0
I have tried this all different ways and keep having problems. I was just shown a new method earlier today and I'm going to try figure that one out this weekend, i'd be interested to find out if there is a method by which this is best solved and it works out well."
Find the solution of the homogenous system.
Add to that particular solutions of your differential equation, use suitable trigonometric trial functions here.
 
If we are just looking for w/w0 to equal near 1, then if we know that the value of omega is sqrt(k/m) and we have k & m, then find that w0 = 4, then we just want w from the forcing term to also be four in order for this to resonate, correct? Am I way off base here? Is this much more difficult than that?

I am working on another word problem then I'm going to redo the other one from scratch using method of undet coeffs.

Thanks!
Angela.
 
You're right, if you hit the natural frequency (given by k&m) with your external force, the system experiences resonance.
 
arildno said:
You're right, if you hit the natural frequency (given by k&m) with your external force, the system experiences resonance.

Wow, I thought it was going to be a much more complicated problem than that! Thanks!


Ok, another one I have a problem with is the one that says "The steady state solution of the diff eq x'' + x' + 2x = cos(t) can be written x(t) = sqrt(1/2)*cos(t + sigma) and asks for sigma where -pi < sigma <+ pi. I haven't a clue on this one, any suggestions? I get yc = e^-t/2[C1cos((sqrt(7)/2)*t + C2sin((sqrt(7)/2)*t] but I'm not sure how this helps.



And, I just worked out this problem : y'' + y = 8cos(2x) - 4sin(x) where y(pi/2) = -1 and y'(pi/2) = 0 using the annihilator method and got :

Ytotal = (16/3 - pi)cos(t) - 11/3*sint - 8/3*cos(2t) + 2tcos(t)

If you get a chance, can you take a look and see if you come out with the same thing because I've gotten different answers with other methods and I'm not sure if I'm doing this one right.

Thanks again for the help!
-A
 
Last edited:

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