I want to parallel transport a vector $V^{\mu}$ with the initial condition $V^{\mu} = (V^{\theta},V^{\phi}) = (1,0)$ along a closed curve parameterized by $ \lambda \in [0,1]$ and determine the resulting vectors dependence on the constant $\theta$.
I've determined that
$$\frac{d V^{\theta}}{d\lambda} - 2\pi \sin\theta \cos\theta \;V^{\phi} = 0 $$ $$\frac{d V^{\phi}}{d\lambda} + 2\pi \cot\theta \; V^{\theta} = 0$$
I'm stuck an this point, I've seen a set of solution and they just substitute the initial condition in for $V^{\phi}$ and $V^{\theta}$, as far as I know this is definitely not how to solve differential equations.
Any insights?
Differentiate the first equation with respect to $\lambda$. You'll obtain a $\frac{d V^\phi}{d\lambda}$ in the second term. Now substitute the $-2\pi \cot\theta \;V^\theta$ for $\frac{d V^\phi}{d\lambda}$ in the first equation. This gives you a second order equation with constant coefficients.