# Inverse Trig Functions as a (unique?) solution to a PDE

1. Mar 20, 2012

Hi,

I know from basic math courses that inverse trig functions are multi valued (e.g. arctan(c)=θ+n*2∏). Now, if I solve a partial differential equation and I get an inverse trig function as part of my solution, does that mean solutions to the pde are non-unique?

For example, if f(x,t)=arctan(t)*sin(x) is a solution to a pde does that mean that the pde doesn't give unique solutions (i.e. no uniqueness)?

The reason I am asking this, is because I found solutions to the 2D navier-stokes equations, where the form of the velocity and pressure functions are arctan(t)*g(x,y). The solutions are 2D and I thought uniqueness had already been proven for 2D. I can provide the solution if you want. Thanks in advance.

2. Mar 21, 2012

### AlephZero

The uniqueness (or not) of the solution to a PDE depends on the boundary conditions.

If the BCs specify a fixed value of the function at some point and the solution is continuous, that effectively makes the arctan function single valued by selecting one mutiple of n*2∏.

Otherwise, there might be multiple solutions.