The solution of Cauchy problem \(xu_x+yu_y=0\) with initial data \(u(x,y)=x\) on \(x^2+y^2=1\) is-
(a) Exist for all \(x\) and \(y\) in \(\mathbb{R}\)
(b) Unique in \(\{(x,y)\in \mathbb{R}^2:(x,y)\neq(0,0)\}\)
(c) Bounded in \(\{(x,y)\in \mathbb{R}^2:(x,y)\ne(0,0)\}\)
(d) Unique in \(\{(x,y)\in \mathbb{R}^2:(x,y)\ne(0,0)\}\) but not bounded.
Explanation: (b) and (c) are correct options.
Tags:
PDE