How should I find the following period ## T ## of this solution?

Math100
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Homework Statement
Consider the differential equation ## \ddot{x}+x+\frac{3}{2}\beta x\lvert x\rvert=0 ##
for ## \beta\geq 0 ## with the solution ## x(t) ## and initial conditions
## x(0)=A>0, \dot{x}(0)=0 ##. Show that the period ## T ## of this solution
is given by ## T=4\int_{0}^{\pi/2}\frac{d\theta}{\sqrt{1+\beta A\sec^2\theta(1-\sin^3\theta)}} ##.
Relevant Equations
None.
From the original equation of ## \ddot{x}+x+\frac{3}{2}\beta x\lvert x\rvert=0 ##,
I've got ## \ddot{x}\dot{x}+x\dot{x}+\frac{3}{2}\beta x\lvert x\rvert\dot{x}=0
\implies \frac{d}{dt}(\frac{1}{2}\dot{x}^2+\frac{1}{2}x^2+\frac{\beta}{2}x^2\lvert x\rvert)=0
\implies \frac{1}{2}\dot{x}^2+\frac{1}{2}x^2+\frac{\beta}{2}x^2\lvert x\rvert=C ## where
## C ## is the constant.

From the work shown above, how should I use the initial conditions of ## x(0)=A>0 ##
and ## \dot{x}(0)=0 ## to find the given period ## T ##? And where does the factor of
## 4 ## come from in front of the integral sign of the period ## T ##?
 
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Set y = \dot x and switch to scaled polar coordinates (x,y) = (Ar \cos \theta, Ar \sin \theta). The period is then given by <br /> T = \int_0^{2\pi} \frac{1}{\dot \theta}\,d\theta. In this case, by symmetry the integral over the full period from 0 to 2\pi is four times the integral over the quarter period from 0 to \pi/2.
 
pasmith said:
Set y = \dot x and switch to scaled polar coordinates (x,y) = (Ar \cos \theta, Ar \sin \theta). The period is then given by <br /> T = \int_0^{2\pi} \frac{1}{\dot \theta}\,d\theta. In this case, by symmetry the integral over the full period from 0 to 2\pi is four times the integral over the quarter period from 0 to \pi/2.
But what's ## \dot\theta ## from the integral?
 
My earlier hint was unhelpful; please disregard it.

Your first integral gives you an expression for |\dot x| in terms of x, <br /> |\dot x| = F(x) where F is even.

You can therefore start from <br /> T = \int_0^T\,dt = \int_A^0 -F(x)^{-1} \,dx + \int_0^{-A} -F(x)^{-1}\,dx + \int_{-A}^0 F(x)^{-1}\,dx + \int_0^A F(x)^{-1}\,dx.
 
Math100 said:
But what's ## \dot\theta ## from the integral?
You can find ## \dot \theta ## by using the substitution ## x=A\sin\theta ##, ## \dot x=A\cos\theta\dot \theta ##.

You do not need to discard the whole post #2. Use the next part
pasmith said:
The period is then given by <br /> T = \int_0^{2\pi} \frac{1}{\dot \theta}\,d\theta. In this case, by symmetry the integral over the full period from 0 to 2\pi is four times the integral over the quarter period from 0 to \pi/2.
to find the expression for ## T ##.
 
Gavran said:
You can find ## \dot \theta ## by using the substitution ## x=A\sin\theta ##, ## \dot x=A\cos\theta\dot \theta ##.

You do not need to discard the whole post #2. Use the next part

to find the expression for ## T ##.
Thank you!
 
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