Hi all! I'm stuck with a system of PDE. I'm not sure I want to write it here in full, so l'll write just one of them. I've found a solution to this equation but I'm not sure it's the most general one since when I plug this solution in to the other eqs, I get a trivility condition for the coefficients [tex] 2\bar{k}^1\left(\bar{s},\bar{t},\bar{u}\right)-2 k^1\left(s,t,u\right)+\left(s-\bar{s}\right)\left(\partial_s k^1\left(s,t,u\right) + \bar{\partial}_{\bar{s}}\bar{k}^1\left(\bar{s}, \bar{t}, \bar{u}\right)\right) =0 [/tex] Can someone help?
Not sure I've understood the equation. Is this simplification valid: 2u(x) - 2v(y) + (y-x)(∂v/∂y + ∂u/∂x) = 0 ? If so: ∂v/∂y + ∂u/∂x = 2(u-v)/(x-y) Consider (u, v+v') is also a solution. So ∂v'/∂y = 2v'/(y-x) v' = (y-x)^{2}f(x) where f is an arbitrary function of x. Does that help?
Thank you for the answer. Ok that was useful, at least a bit. In fact, although it is correct, I need a u and a v depending ONLY from one of the two variables (x and y). In fact I'm dealing with (anti)holomorphic functions and I need them to respect the holomorphicity condition.