What is this differential equation? I'm going crazy

[SSGD]

I have been working on a math problem and I keep getting the some type of PDEs.

x*dU/dx+y*dU/dy = 0
x*dU/dx+y*dU/dy+z*dU/dz = 0 ...

x1*dU/dx1+x2*dU/dx2+x3*dU/dx3 + ... + xn*dU/dxn= 0

dU/dxi is the partial derivative with respect to the ith variable. Does anyone know about this type of PDE? It looks like the Euler-Cauchy ODE.

 

[jedishrfu]

Here's something on Euler-Cauchy ODE:

http://tutorial.math.lamar.edu/Classes/DE/EulerEquations.aspx

It doesn't look like it fits the bill though, but I'm pretty rusty with these things now. The Euler Cauchy shows ax^2 y'' + bx y' + c y = 0

 

[fzero]

In each case, we can write the differential operator as $\vec{r}\cdot\nabla_r U$, so the operator is a directional derivative along the vector $\vec{r}$. I'm not sure that the equation with this equal to zero has a special name, though there are equations named after both Euler and Cauchy in the theory of transport that involve a directional derivative.

 

[SSGD]

The equations can be reduced to a linear pde with constant coefficients with the substitution.

xi = e^yi

dU/dy1+dU/dy2 + ... + dU/dyn= 0

Same substition you would use to solve Euler-Cauchy ODE.

 

[fzero]

You can actually do a bit more. On a patch where one coordinate does not vanish, $x_n\neq 0$, we can show that the solutions $U(x_i)$ must only depend on the ratios $x_i/x_n$ (called homogenous or projective coordinates), i.e., $U$ is a function $U(x_i/x_n)$.

 

[SSGD]

Wow your right... So U(xi/xn) could be written as U(e^(y1-yn),e^(y2-yn),...,e^(yn-1-yn)).

 

[SSGD]

The solution U(x1,x2,...,xn)=U(a*x1,a*x2,...,a*xn)

Where a is constant. So it doesn't scale. I was just making substitutions to look for properties.

... Just realized the ratio of the two variables would cancel any constant coefficients anyway ... A lot of work for the same result.

Now I need a condition to find a solution.

 

[fzero]

Any (first-differentiable) function of the homogenous variables is a solution. You would need additional differential equations or boundary conditions to narrow the solutions.

 

[SSGD]

I know there has to be n-1 solutions for the solution I am looking for if they exist. So one less than the number of variables. Which I think is possible because you can rewrite the function as the ratio of one of the variables. Which makes one less than n inputs. Thanks for the help/enlightenment. I like the vector approach. I am going to write out the Fourier series for the solution, sub back in the ln(yi) for xi and see if I can find some conditions

 

[pasmith]

I have been working on a math problem and I keep getting the some type of PDEs.

x*dU/dx+y*dU/dy = 0
x*dU/dx+y*dU/dy+z*dU/dz = 0 ...

x1*dU/dx1+x2*dU/dx2+x3*dU/dx3 + ... + xn*dU/dxn= 0

dU/dxi is the partial derivative with respect to the ith variable. Does anyone know about this type of PDE? It looks like the Euler-Cauchy ODE.

Those are first-order linear PDEs which can in principle be solved by the method of characteristics.

As an example, if $$x \frac{\partial U}{\partial x} + y\frac{\partial U}{\partial y} = 0$$ then you can make a change of independent variables from $(x,y)$ to $(t,s)$ where $\frac{\partial x}{\partial t} = x$, $\frac{\partial y}{\partial t} = y$ to obtain $$\frac{\partial U}{\partial t} = 0.$$ Thus $U = f(s)$ for some function $f$.
Now we know that $(x,y) = (A(s),B(s))e^t$ so all we need is to choose the functions $A$ and $B$ appropriately and we can then invert the relationship to give $(t,s)$ in terms of $(x,y)$. Here we can take $(A(s),B(s)) = (\cos s, \sin s)$ which gives a problem at $(x,y) = (0,0)$, but if you look at the PDE you see that $\nabla U$ is not uniquely determined at (0,0) anyway.