Parallel Transport on a sphere

[Mike706]

Homework Statement

See attached picture of the problem.

Homework Equations

See attached picture of the pair of diff equations.

The Attempt at a Solution

I was able to solve the problem up to the point the picture gets to. However, the author says he obtained this by integrating. I couldn't figure out how to do this by direct integration, so I modified the pair of equations into two 2nd order ODE's, each of one function, and solved them.

Is there a way to directly integrate these equations? I have a feeling I'm going to feel dumb when I hear the answer. Please let me know if you need any more info.

Thanks for your help,
Mike

 

[mighty2000]

Hello Mike,

Thank you for sharing your attempt at solving this problem. It seems like you have made good progress so far. To directly integrate these equations, you will need to use the method of separation of variables. This method involves isolating the variables on opposite sides of the equation and integrating each side separately.

For the first equation, you can write it as:

dy/dx = (1/2)y

To solve this, you can separate the variables by dividing both sides by y and multiplying both sides by dx:

dy/y = (1/2)dx

Integrating both sides will give you:

ln|y| = (1/2)x + C

Where C is a constant of integration. You can then solve for y by taking the exponential of both sides:

y = e^(1/2)x + C

You can follow a similar process for the second equation and solve for z. I hope this helps. Let me know if you have any further questions.