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Here's an expansion of my comment that the natural formulation of this problem as an EDS is on the coframe bundle $\pi: P\to M$, which, I hope, will be helpful. Also, because it will match my usual n …

Your equations are equivalent to the $1$-form equations $$ \mathrm{d}f = X_0(f,g)\,\mathrm{d}s + Y_0(f,g)\,\mathrm{d}t \quad \text{and}\quad \mathrm{d}g = X_1(f,g)\,\mathrm{d}s + Y_1(f,g)\,\mathrm{ …

In their comments to my first answer, the OP has clarified that they did not mean to regard the metric $h$ as a given, but, rather, an output of the problem of prescribing coframings by specifying the …

Let me phrase the problem as I understand the given data and then describe how the 'theory of exterior differential systems' would be applied. One starts with a compact Riemannian $3$-manifold $(M,h)$ …

I thought about your problem and realized that there is no coordinate parametrization of the catenoid with the properties that you want. Here is my argument: First, note that, in the given $uv$-param …

You are correct that Cauchy-Kovalevskaya does not apply directly to this problem, but there are other theorems that give sufficient conditions, provided that you make certain basic regularity assumpti …

Yes, this can be integrated explicitly. First, notice that, since $m\not=0$, we can write $\alpha(t) = 2m\bigl(x(t)+iy(t)\bigr)$, in which case, the given equation becomes $$ \dot x + i\,\dot y = -4( …

The general solution of your equations in a simply connected domain on which $r_2\not=0$ and $r_1\not=\pm1$ is $$ \beta = \frac12 + \frac1{{(r_1}^2{-}1)}\, \left(\frac{\partial a}{\partial\theta_1}+b( …

Generally, you want there to be a non-characteristic transversal, i.e., a (let's say, smooth) curve $C$ in your domain $D$ such that each segment of each line $x=x_0$ in $D$ is connected and meets $C$ …

Since everything is local and $C^\infty$, it is not hard to derive sufficient conditions for there to exist solutions. Analyticity is not actually needed, but some assumption of regularity is necessa …

In dimension $1$, it's true that a flowboxing change of coordinates for a $C^r$ vector field is $C^{r+1}$, but this is no longer true in dimensions greater than $1$. Basically, the reason is this: If …

Remark: I had a little time to write a draft of my notes on the proofs of the claims I make below and have posted it on my home webpage here. (It would have made a very long post on MO, so I decided …

Well, here are a few comments that might be helpful: First, if one sets $q = a/R>0$, then the equation the OP wants to study can be written in the form $$ \dot\theta^2 +\omega^2\,\sin^2\theta - q^2\,t …

As I understand it, the equation you are imposing on the function $\theta(x,y)$, defined on a domain $D\subset\mathbb{R}^2$ in the $xy$-plane is that, for some positive constants $\lambda_1\not=\lambd …

Note that, if you take $\phi=0$, then the equation reduces to $w_y =0$, i.e., if $D\subset C$ is the domain of $w$ and $x:D\to\mathbb{R}$ is the projection on the $x$-axis and has connected fibers, t …