Math100
- 813
- 229
- 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 ##?
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 ##?