# How to solve this vibration equation?

• I
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.

Mentor
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.

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dlgoff
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?

Mentor
2021 Award
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.

Delta2 and jedishrfu
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?

Mentor
2021 Award
... 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.

Mentor
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?

jedishrfu
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
Homework Helper
Gold Member
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.