How Is the Solution to the Vibration Equation Derived?

[thaiqi]

TL;DR Summary

How to solve this vibration equation?
$$\ddot{x} + \omega_{0}^2 x = {e \over m} E_{x}$$

Hello,everyone:
I got an equation:
$$\ddot{x} + \omega_{0}^2 x = {e \over m} E_{x}$$
I know the solution is:
$$x(t) = {e \over {\omega_{0} m}} \int_{0}^{t} E_{x}(\xi) \sin{ \omega_{0} } (t - \xi) d \xi \\$$
$$x(0) = \dot{x} (0) = 0$$

My attempt to verify this solution is by using formula:
$${d \over dx } \int_{\alpha (x)}^{\beta (x)} f(x,y)dy = \int_{\alpha (x)}^{\beta (x)} { {\partial f(x,y)} \over {\partial x}} dy + f[x, \beta (x)] \beta^{\prime}(x) - f[x, \alpha (x)] \alpha^{\prime}(x)$$
But I don't know how the physicist obtained it. Can anyone give an answer? Thanks.

 

[jedishrfu]

We don't give answers. You need to show what attempt you made to solve it and I don't see that in your post. We in turn will give you hints or tell you where you went wrong. This is how our site works.

We also have a template to fill out for homework type questions when posted into one of our homework forums.

1) Homework statement

2) Relevant equations

3) Attempt at a solution

 

[thaiqi]

We don't give answers. You need to show what attempt you made to solve it and I don't see that in your post. This is how our site works.

Thanks. Is there any book talking about equations of this form?

 

[jedishrfu]

There's Khan Academy or MathIsPower4u which have online tutorials on differential equations.

http://www.mathispower4u.com/diff-eq.php

Introduction to Free Undamped Motion (Spring System)
Ex 1: Free Undamped Motion IVP Problem (Spring System)
Ex 2: Free Undamped Motion IVP Problem (Spring System)
Ex 3: Free Undamped Motion IVP Problem (Spring System)
Ex: Model Free Damped Vibration and Find Displacement Function
Ex: Determine a Dampening Force For An Overdamped System (Free Damped Vibration)
Ex: Determine a Dampening Force For An Critically Damped System (Free Damped Vibration)

The differential equation you have is for simple harmonic motion.

https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Simple_Harmonic_Motion

 

[thaiqi]

There's Khan Academy or MathIsPower4u which have online tutorials on differential equations.

http://www.mathispower4u.com/diff-eq.php

……

Thank you.

 

[thaiqi]

There's Khan Academy or MathIsPower4u which have online tutorials on differential equations.

http://www.mathispower4u.com/diff-eq.php

Introduction to Free Undamped Motion (Spring System)
Ex 1: Free Undamped Motion IVP Problem (Spring System)
Ex 2: Free Undamped Motion IVP Problem (Spring System)
Ex 3: Free Undamped Motion IVP Problem (Spring System)
Ex: Model Free Damped Vibration and Find Displacement Function
Ex: Determine a Dampening Force For An Overdamped System (Free Damped Vibration)
Ex: Determine a Dampening Force For An Critically Damped System (Free Damped Vibration)

The differential equation you have is for simple harmonic motion.

https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Simple_Harmonic_Motion

Sorry I can't visit them in my country. Could you please suggest any other books?

 

[fresh_42]

Summary:: How to solve this vibration equation?
$$\ddot{x} + \omega_{0}^2 x = {e \over m} E_{x}$$

ut I don't know how the physicist obtained it. Can anyone give an answer? Thanks.

You basically have an equation of the form $x''(t)+cx(t)=d$ which is a simple second order differential equation. There are a lot of techniques to solve differential equations in general, but this one is especially simple and occurs very often in physics. You can e.g. set $x(t))=a \sin(\alpha t)+b\cos(\beta t)$ and solve the equation system which results from differentiating. $x'' \sim x$ as given, already looks like sine and cosine, since those functions "reproduce" themselves by differentiation twice.

 

[thaiqi]

You basically have an equation of the form $x''(t)+cx(t)=d$ which is a simple second order differential equation. There are a lot of techniques to solve differential equations in general, but this one is especially simple and occurs very often in physics. You can e.g. set $x(t))=a \sin(\alpha t)+b\cos(\beta t)$ and solve the equation system which results from differentiating. $x'' \sim x$ as given, already looks like sine and cosine, since those functions "reproduce" themselves by differentiation twice.

Thanks first. It is rather of $x''(t)+cx(t)=d(t)$ type.
Second, I learned from some articles that this can be solved using what is called "Duhamel integral", will a common book on ordinary differential equation talk about that? If not, what book is suitable?

 

[fresh_42]

... will a common book on ordinary differential equation talk about that? If not, what book is suitable?

Not sure whether it will talk about that integral, but certainly about equations of this form. A standard textbook about differential equations is a good recommendation for any physicist. However, I cannot name a special one.

 

[jedishrfu]

Some folks use Arfken and Weber Mathematical Methods for Physicists or Boaz.

Theres also Nearings book available free online.

http://www.physics.miami.edu/~nearing/mathmethods/

not sure if duhamel is mentioned though. You could search on diff equation books duhamel to find one since google indexed so many so far.

 

[mpresic3]

You are starting with a second order differential equation.
That rule you wrote involving the integral below is called Leibniz's rule. It is in the mathematical handbook by Schaums. The physicist (probably) attained it when (s)he was taking Calculus or perhaps advanced calculus classes. The rule (and even the method of solution) is also found in many books on modern control theory in engineering. Electrical and Aerospace engineers are well acquainted with these types of problems; but allow me to be of even more help.
Your starting point is a second order differential equation, and Leibniz rule involves an (i.e. one) integral. In order to use it you should express your second order differential equation as a first order differential equation involving matrices. Then you would obtain the state transition matrix. etc.
Your best bet would be to read relevant sections in Modern Control theory such as Modern Control Theory by Brogan. If you are a physicist, this book (and others in this area), are likely unfamiliar to you, and I do not know of the references to help.

 

[thaiqi]

Thanks for you all.

 

[etropy]

Sorry for the kinda stupid qustion, but is $$E_x$$ a function E differentiated with respect to x, or is $$E_x$$ just a constant?

 

[peterwang]

Looking at the form of the equation, it involves time t, so Laplace transformation with respect to time t would be helpful. Looking at the form of the solution, it is the Laplace transformation of a convolution. So Laplace transformation is the solution.

 

[Delta2]

Laplace transform is surely one way to go. Also (maybe not sure) Fourier transform. But i think it can be solved also with the Green's function method because the ODE is of the form $$Lf(t)=s(t)$$ where L a linear differential operator and s(t) a source function.