Differential Equation - Determining frequency of beats/rapid oscillations

In summary, the given equation can be solved using the characteristic equation and finding the general solution. The natural period and frequency can be calculated using the coefficients from the general solution. To determine the frequency of the beats, the interaction between the functions in the homogeneous solution and the particular solution must be considered. To find the frequency of the rapid oscillations, the imposed long wave (low frequency) on a short wave (high frequency) must be taken into account.
  • #1
cse63146
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]
 
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  • #2
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
cse63146 said:

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
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]
 

1. What is a differential equation?

A differential equation is a mathematical equation that represents the relationship between a function and its derivatives. It is used to describe how a function changes over time or in response to other variables.

2. How is frequency of beats determined using differential equations?

The frequency of beats can be determined using the equation f_beat = |f_1 - f_2|, where f_1 and f_2 are the frequencies of the two waves that are causing the beats. This equation can be derived from a differential equation that models the superposition of two waves with slightly different frequencies.

3. What is the significance of determining the frequency of beats?

Determining the frequency of beats is important in many fields, such as acoustics, music, and engineering. It can help in tuning musical instruments, studying the properties of sound waves, and designing electronic circuits.

4. How does the presence of rapid oscillations affect the determination of beat frequency?

If there are rapid oscillations present in the waves, it can make it difficult to accurately determine the beat frequency. In this case, a more sophisticated approach, such as Fourier analysis, may be needed to accurately determine the frequency of beats.

5. Can differential equations be used to model other phenomena besides beats?

Yes, differential equations are widely used in many areas of science and engineering to model a variety of phenomena, such as population growth, heat transfer, and fluid dynamics. They are a powerful tool for understanding and predicting the behavior of complex systems.

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