Two Variable PDE in Open Domain Plane

[lavinia]

In an open domain in the plane

(xU$_{x}$-yU$_{y}$-U)/U =

(xV$_{x}$ - y V$_{y}$+V)/V

 

[TylerH]

I'd like to know the answer myself. Does $\frac{xU_x}{U}-\frac{xV_x}{V}=\frac{yU_y}{U}-\frac{yV_y}{V}$ make it any easier? I've only dabbled in PDEs myself, but that seems like a separation of variables problem, when rephrased like that.

EDIT: Mathematica gives an error saying the answer is indeterminate, because there are more dependent variables than equations.

 

[lavinia]

I'd like to know the answer myself. Does $\frac{xU_x}{U}-\frac{xV_x}{V}=\frac{yU_y}{U}-\frac{yV_y}{V}$ make it any easier? I've only dabbled in PDEs myself, but that seems like a separation of variables problem, when rephrased like that.

EDIT: Mathematica gives an error saying the answer is indeterminate, because there are more dependent variables than equations.

U = x V = K/x sort of works but doesn't allow the origin or the y axis.
Also the answer is a little trivial.

 

[MathematicalPhysicist]

So you have:
x d/dx ln (U) -y d/dy ln(U) -1 = xd/dx ln (V) -y d/dy ln(V) +1
x d/dx (ln(U/V))=y d/dy (ln(U/V))+2

Now make a guess of a function of the type U/V = exp(F(x,y))

to get:
x d/dx F = y d/dy F +2

so you have x d/dx F -y d/dy F =2

d/dx (xF) - d/dy (yF) = 2

Don't see how to solve this, though.

p.s the derivatives above are partial btw.

 

[Ben Niehoff]

In an open domain in the plane

(xU$_{x}$-yU$_{y}$-U)/U =

(xV$_{x}$ - y V$_{y}$+V)/V

I would rewrite this as

$$x \partial_x (\log U) - y \partial_y (\log U) - 1 = x \partial_x (\log V) - y \partial_y (\log V) + 1$$

or

$$x \partial_x (\log U - \log V) - y \partial_y (\log U - \log V) = 2$$

Define some function $e^f = U/V$ and you have

$$x \partial_x f - y \partial_y f = 2$$

which should be easy to solve by characteristics. Mathematica gives

$$f = 2 \log x + C x y$$

for arbitrary C. Then U and V can be any functions satisfying

$$\frac{U}{V} = x^2 e^{C x y}$$

 

[lavinia]

Thanks Ben

What approach would you recommend to learning more about differential equations? Right now i am learning differential geometry.

 

[Ben Niehoff]

I don't know, I taught myself the method of characteristics when I started running across problems in my research that needed it.

 

[MathematicalPhysicist]

Sometime I wonder why I even bother answering. :-/

 

[lavinia]

I thank you for this solution and feel that I should explain where this equation came from.

To learn some differential geometry I posed the question of what vector fields on a surface can be tangent to geodesics for some Riemannian metric. Computation led to the condition that a unit length vector field V is tangent to geodesics if its Lie bracket with iV, the unit vector field orthogonal to it, is a multiple of iV (and that this multiple satisfies a differential equation along the geodesic that relates it to the Gauss curvature of the metric).

For the vector field xd/dx - yd/dy in a neighborhood of the origin in the plane, one gets the differential equation in this post. I think your solution shows that the field of hyperbolas y = k/x can be geodesics and that iV can be chosen to be x$^{2}$d/dx + e$^{-xy}$d/dy.

So it seems that a vector field of index -1 around a singularity can be a geodesic flow.

I think the Gauss curvature is -1 for this example but I haven't yet done the ugly calculation.