Forced Damped Pendulum (but really just nonhomog?) due tmw..

[blubby]

1) Find the general solution to: d2u/dt2 + du/dt + u = 3*sin(sigma*t)

2) For 1, find sigma where there is no resonance

3) Would 2 change if 3*sin(sigma*t) were changed to 10*sin(sigma*t)?


Okay, so my main question is on #1, I hope I can figure out the rest once I have the general solution. I tried to solve this like previous homogenous/nonhomogenous equations. Here we go:

d2u/dt2 + du/dt + u = 3*sin(sigma*t)
d2u/dt2 + du/dt + u = 0
u_h = [e^(-1/2t)]*[cos(rt3)/2*t + sin (rt3)/2*t)


so then to find a particular solution, I used u_p = A*cos(sigma*t) + B*sin(sigma*t) and took first and second derivatives of that and then substituted into the original equation to get:

cos(sigma*t)[-A*(sigma^2) + B*(sigma) +A] + sin(sigma*t)[-B*(sigma^2) - A*(sigma) +B] = 3*sin(sigma*t)


I have no idea where to go from here because if I try to isolate A's and B's or isolate cos's and sin's, I just can't solve this. Am I approaching this wrong? Any help much appreciated!

--blubby

 

[akoaysigod]

Set the coefficients on the left equal to the ones on the right, ie,

-As^2 + Bs + A = 0
-Bs^2 - As + B = 3

I think is what you have, then solve for A and B, treating sigma as a constant. Then plug the result for A and B back in and you should find the correct answer, it's a bit tedious but you were on the right track. At least, I'm pretty sure that's what you have to do.