# I Electrical circuit differential equation

1. May 13, 2017

### Polygon

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.

2. May 13, 2017

### BvU

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 $100 \sin(-\omega t)$ ?

3. May 13, 2017

### Polygon

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?

4. May 13, 2017

### BvU

(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.

5. May 13, 2017

### Polygon

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.

6. May 13, 2017

### BvU

Does that satisfy the differential equation ? I don't think it does...

7. May 13, 2017

### Polygon

It does after solving for the constants. Thanks for your time BvU. I think I've got it now :)

8. May 14, 2017

### BvU

Also for $\omega\ne \omega_0$ ?

And: you're welcome.