Electrical circuit differential equation

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Discussion Overview

The discussion revolves around the differential equation of an electrical circuit, specifically focusing on the conditions for resonance. Participants explore the mathematical solutions to the equation and the implications of angular frequency values.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes that resonance occurs when ω=√20 and questions whether -√20 should also be considered, seeking clarification on the significance of negative angular frequency.
  • Another participant challenges the completeness of the solution and questions the role of constants A and B in the context of the homogeneous solution.
  • A later reply suggests that the difference between sine terms with positive and negative frequencies may not significantly affect the overall solution due to the arbitrary nature of the constants involved.
  • There is a discussion about the general solution structure, with one participant asserting that it should include both a particular and a homogeneous solution, while another expresses uncertainty about the correctness of the proposed form.
  • One participant expresses a need for confirmation regarding the resonance conditions and acknowledges their limited knowledge of electrical circuits.
  • Another participant raises a question about the validity of the proposed general solution in satisfying the differential equation.
  • There is a mention of the conditions for ω not equal to ω₀, indicating further exploration of resonance conditions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the completeness of the solution or the implications of negative angular frequency. Multiple competing views remain regarding the formulation of the general solution and the resonance conditions.

Contextual Notes

Some participants express uncertainty about the mathematical steps involved in deriving the general solution and the role of integration constants. The discussion reflects varying levels of familiarity with the topic.

Who May Find This Useful

Individuals interested in electrical circuit theory, differential equations, and resonance phenomena may find this discussion relevant.

Polygon
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q''+ 20 q = 100 sin(ωt)

I have been asked to find all mathematically possible values of ω for which resonance will occur. From the homogeneous solution, q(t) = Acos(√20 t) +Bsin(√20 t), I can see that resonance occurs when ω=√20. My question is, should I also consider -√20? And if so, what is the significance of a negative angular frequency?

Thanks.
 
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You sure you have the full solution ?
What determines A and B ?
Is ##C\cos(-\sqrt{20} \, t) + D\sin(-\sqrt{20} \,t )\ ## much different from ##A\cos(\sqrt{20} \, t)+ B\sin(\sqrt{20} \,t ) \ ## ?

Likewise: what's the difference between ##100 \sin(\omega t)## and ##100 \sin(-\omega t)## ?
 
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BvU said:
You sure you have the full solution ?
What determines A and B ?
Is ##C\cos(-\sqrt{20} \, t) + D\sin(-\sqrt{20} \,t )\ ## much different from ##A\cos(\sqrt{20} \, t)+ B\sin(\sqrt{20} \,t ) \ ## ?

Likewise: what's the difference between ##100 \sin(\omega t)## and ##100 \sin(-\omega t)## ?

Thank you for your reply BvU. The only difference is that the sine term will have a negative in front, which doesn't really make a difference since A, B, C and D are arbitrary constants. So my particular solution in both cases (ω=+√20 and ω=-√20 ) will be of the form q(t) = t(acos(√20 t) ± bsin(√20 t)), which means the charge will oscillate with increasing amplitude and without bound as t increases. So these are the two (and only two) ω values for which resonance occurs in this system?
 
Polygon said:
$$q(t) = t(A \cos(√20 t) ± B\sin(√20 t))$$
(If that is what you mean) Doesn't look correct.
General solution = one particular solution + solution of homogeneous equation ,
the first without integration constants, the second with​
is what I remember.
 
Sorry, that is not what I meant. I should not have written it like that. As you said, the general solution contains a particular solution and the homogeneous solution. So the general solution will be of the form q(t) = Acos(√20 t) +Bsin(√20 t) + t(acos(√20 t) + bsin(√20 t)).

So the values of ω that will give resonance are both +√20 and -√20, right? I just need some confirmation as I haven't been given a problem like this before and I have little knowledge of electrical circuits.
 
Polygon said:
q(t) = Acos(√20 t) +Bsin(√20 t) + t(acos(√20 t) + bsin(√20 t)).
Does that satisfy the differential equation ? I don't think it does...
 
BvU said:
Does that satisfy the differential equation ? I don't think it does...

It does after solving for the constants. Thanks for your time BvU. I think I've got it now :)
 
Also for ##\omega\ne \omega_0## ?

And: you're welcome.
 

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