- #1

Math100

- 771

- 219

- Homework Statement
- Find the equilibrium points and the general equation for the phase paths of ## \ddot{x}+cos(x)=0 ##. Obtain the equation of the phase path joining two adjacent saddles.

- Relevant Equations
- For the general equation ## \ddot{x}=f(x, \dot{x}) ##, equilibrium points lie on the ## x ## axis, and are given by all solutions of ## f(x, 0)=0 ##, and the phase paths in the plane ## (x, y) (y=\dot{x}) ## are given by all solutions of the first-order equation ## \frac{dy}{dx}=\frac{f(x, y)}{y} ##.

Consider the differential equation ## \ddot{x}+cos(x)=0 ##.

Note that ## \ddot{x}=f(x, \dot{x}) ##, so we have ## f(x, y)=-cos(x) ##.

Then ## f(x, 0)=-cos(x)=0 ##.

This gives ## x=n\pi-\frac{\pi}{2} ## for some ## n\in\mathbb{Z} ##.

Since the differential equation for the phase paths is given by ## \frac{dy}{dx}=-\frac{cos(x)}{y} ##,

it follows that ## y dy=-cos(x)dx\implies \int y dy=-\int cos(x)dx\implies \frac{y^2}{2}=-sin(x)+C ##.

Thus, ## y^2=-2sin(x)+C\implies y=\pm\sqrt{C-2sin(x)} ## where ## C ## is an arbitrary constant.

Above is my work for this problem. However, I've only found the general equation/solution for the phase paths of ## \ddot{x}+cos(x)=0 ## but how should I find the equilibrium points based on the equation I've found above in my work ## x=n\pi-\frac{\pi}{2} ## for some ## n\in\mathbb{Z} ##? Also, how should I obtain the equation of the phase path joining two adjacent saddles from here?

Note that ## \ddot{x}=f(x, \dot{x}) ##, so we have ## f(x, y)=-cos(x) ##.

Then ## f(x, 0)=-cos(x)=0 ##.

This gives ## x=n\pi-\frac{\pi}{2} ## for some ## n\in\mathbb{Z} ##.

Since the differential equation for the phase paths is given by ## \frac{dy}{dx}=-\frac{cos(x)}{y} ##,

it follows that ## y dy=-cos(x)dx\implies \int y dy=-\int cos(x)dx\implies \frac{y^2}{2}=-sin(x)+C ##.

Thus, ## y^2=-2sin(x)+C\implies y=\pm\sqrt{C-2sin(x)} ## where ## C ## is an arbitrary constant.

Above is my work for this problem. However, I've only found the general equation/solution for the phase paths of ## \ddot{x}+cos(x)=0 ## but how should I find the equilibrium points based on the equation I've found above in my work ## x=n\pi-\frac{\pi}{2} ## for some ## n\in\mathbb{Z} ##? Also, how should I obtain the equation of the phase path joining two adjacent saddles from here?