• Support PF! Buy your school textbooks, materials and every day products Here!

Differential Equation - Determining frequency of beats/rapid oscillations

  • Thread starter cse63146
  • Start date
  • #1
452
0

Homework Statement



y'' + 6y = 2cos3t

a)Determine frequency of the beats
b)Determine frequency of the rapid oscillations
c)Use the information from parts a) and b) to give a rough sketch of a typical solution

Homework Equations





The Attempt at a Solution



Not sure how to do the first 2 parts. I know the natural period is [tex]\frac{2 \pi}{B}[/tex] and natural frequency is [tex]\frac{B}{2 \pi}[/tex]
 

Answers and Replies

  • #2
33,152
4,838
The characteristic equation of the homogeneous equation is r2 + 6 = 0, so its solutions are r = +/- i[itex]\sqrt{6}[/itex]. This means that
[tex]y_h = Ae^{i\sqrt{6}t} + Be^{-i\sqrt{6}t}[/tex]
It's more convenient to write a different linear combination of these to get
[tex]y_h = c_1cos(\sqrt{6}t) + c_2sin(\sqrt{6}t)[/tex]

For the nonhomogeneous equation, look for a solution yp = Acos(3t) + Bsin(3t).

The general solution will be yh + yp.

For a, you'll have to figure out how the functions in the homogeneous solution interact with what I'm pretty sure will be the one particular solution function, and how often the waves of the two periodic parts reinforce each other (the beats).

For b, you need to find the rapid oscillation frequency. You have in essence a long wave (low frequency) imposed on a short wave (high frequency).

Hope that helps. It's getting late where I am, so I'm going to turn in.
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,791
919

Homework Statement



y'' + 6y = 2cos3t

a)Determine frequency of the beats
b)Determine frequency of the rapid oscillations
c)Use the information from parts a) and b) to give a rough sketch of a typical solution

Homework Equations





The Attempt at a Solution



Not sure how to do the first 2 parts. I know the natural period is [tex]\frac{2 \pi}{B}[/tex] and natural frequency is [tex]\frac{B}{2 \pi}[/tex]
Since there is no "B" in the statement of your problem, that makes no sense! Mark44 has already pointed out that the general solution to the associated homogeneous equation is [itex]A cos(\sqrt{6}t)+ B sin(\sqrt{6}t)[/itex]. That tells you that the natural period is [itex]2\pi/\sqrt{6}[/itex].

Trig identity: cos(s+ t)= cos(s)cos(t)- sin(s)sin(t) so if you have a formula of the form Acos(2\pi t/\sqrt{6})+ Bcos(3t) you can write that, by careful manipulation of A and B, as [itex]cos((2\pi/\sqrt{6}+ 3)t)[/itex].
 
  • #4
452
0
It's not a B. It's supposed to be the greek uppercase Beta.

as in - [itex]A cos(Bt)+ B sin(Bt)[/itex]. It's how my textbook/prof write it.

In this case B (Beta) = [itex]\sqrt{6}[/itex]
 

Related Threads for: Differential Equation - Determining frequency of beats/rapid oscillations

  • Last Post
Replies
3
Views
923
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
0
Views
999
Replies
1
Views
1K
Replies
10
Views
3K
  • Last Post
Replies
3
Views
1K
Top