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Differential Equations Help

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Posted · Original PosterOP

How do you find the frequency of an oscillation from the characteristic equation without solving the whole diffeq? 

b = 0

c = 4 pi2

z+ bz + c = 0, z = +- 2i Pi


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4 minutes ago, wii8cookies said:

2+2=4

Are you positive? Let me count using my fingers.


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Posted · Original PosterOP
3 minutes ago, DoctorZeus said:

at least post a picture of the problem

There's no need for a picture of the problem. It's actually a very simple question for anyone who has taken a Differential Equations class. 


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Posted · Original PosterOP
2 minutes ago, DoctorZeus said:

It's so simple? You solved it then! Yay!!

It's very simple for someone who has taken a diffeq class. I haven't. This is akin to asking how do you find the solutions to something like this: y^2 + 2y + 3 = 0. You just need to know about the quadratic formula and then it's easy. 


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13 minutes ago, wii8cookies said:

2+2=4

Your calculator is broken!

My is working properly.

1+1=0


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i tried, but I didn't understand anything of what you said except for the numbers so I wasn't sure what you are looking for. Im just a high school student with education up to alg2, currently studying trig. This is all I was able to come up with. sorry i couldn't help. good luck tho

IMG_3965.PNG


 

;(

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The imaginary part of the root of the auxiliary equation is the circular frequency, and the real part is the coefficient for the exponential.

You basically just have to convert the root from rads/s to revs/s (Hz).


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7 hours ago, djdwosk97 said:

It's very simple for someone who has taken a diffeq class. I haven't. This is akin to asking how do you find the solutions to something like this: y^2 + 2y + 3 = 0. You just need to know about the quadratic formula and then it's easy. 

Is there really a quadratic formula for *everything? * god damn, and here I am being stumped by adding quad. fractions


- snip-

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7 hours ago, SCGazelle said:

i tried, but I didn't understand anything of what you said except for the numbers so I wasn't sure what you are looking for. Im just a high school student with education up to alg2, currently studying trig. This is all I was able to come up with. sorry i couldn't help. good luck tho

IMG_3965.PNG

My brain bits, they're everywhere. 


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33 minutes ago, MrDynamicMan said:

Is there really a quadratic formula for *everything?* god damn, and here I am being stumped by adding quad. fractions.

In this instance he actually is directly solving a quadratic equation.


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Posted · Original PosterOP
1 hour ago, Dash Lambda said:

 

You basically just have to convert the root from rads/s to revs/s (Hz).

Perfect, thanks.


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I have studied differential equations almost up to a college undergrad level, and I don't understand your question at all o.O By characteristic equation of an oscillation, do you mean some sort of periodic waveform , like y=A cos( ω t ), or something more general ? If you give me an individual question, I can tell you how to solve  it and how to find certain values using shortcuts without solving the differential equation.... If you are referring to questions where you can find the angular frequency  ω for a general equation of oscillation, Then  ω =2 π f, where all the symbols have their usual meanings. If it's an equation of SHM ( Simple Harmonic Motion ), then you can find the value of ω using the below formula and then relate ω and f. ω = Angular frequency, A = Amplitude of oscillation, t = time, f = frequency.

  Image result for differential equation of SHM-SHM equation

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Posted · Original PosterOP

@Kumaresh

 

I figured it out thanks to @Dash Lambda 

 

But you can use the characteristic equation in order to solve oscillating diffeq's.

Given: y''[t] + by'[t] + cy[t] = 0, y[0] = x, y'[0] = z

 

Then: 

Characteristic equation  = z^2 + bz + c = 0. (solve for z-> z1, z2) 

GenSol[t] = k1/e^(z1*t) + k2/e^(z2*t) 

GenSol'[t] = z1*t * k1/e^(z1*t) + z2*t * k2/e^(z2*t)

Set gensol[0] = y[0] and gensol'[0] = y'[0] and solve for k1 and k2. 

 

Plug k1/k2 back into gensol[t] and then you can plot that as your oscillating solution to the diffeq. So you can see the frequency in the plot, or you can find the frequency from the characteristic equation. So, according to @Dash Lambda if z = 2Pi i, then the frequency would be 2Pi * 1 Rev/2Pi = 1.


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6 hours ago, MrDynamicMan said:

Is there really a quadratic formula for *everything? * god damn, and here I am being stumped by adding quad. fractions

No! There isn't! For example, there is no known equation to find the roots of a 5th degree polynomial. 


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Looks like the D.E of a damped oscillation.... That is above my area of speciality :/ I don't know all the special forms required to solve these second order D.E yet.....

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Posted · Original PosterOP
13 minutes ago, Kumaresh said:

@djdwosk97 The derivative of gensol[t] looks wrong. Check it again.....

I misstyped typed it, but I wrote it down correctly when I did it. 

 

GenSol'[t] = (-z1 * k1/e^(z1*t)) - (z2 * k2/e^(z2*t))


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9 hours ago, Kumaresh said:

Looks like the DE of a damped oscillation... That is above my area of speciality :/ I don't know all the special forms required to solve these second order DE yet...

It's actually undamped. He gives the general form of a damped oscillation, but the damping coefficient is 0.

 

The characteristic equation of a homogeneous LDE works by assuming a solution can be expressed in the form y = Cert, and since y(n) = rny, you can substitute rn for y(n) to find r. (He used z.)

 

Per this form, if the root is a real number, the solution is an exponential. If it's complex, the solution is y = Ce(a + bi)t = Ceate(bi)t, which by Euler's identity can be expressed as y = eat(C1sin(bt) + C2cos(bt)).

 

Therefore, if the real part is 0, as is the case in x2 + c = 0 (where c is a positive constant), eat = 1 and the motion is undamped.


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