If you think about the problem geometrically
the left hand side is the directional derviative of the function.
This states the function is constant in the direction of the vector
This tells us that
This is a family of hyperbola's not circles as the problem asks......
Your PDE needs to have one of the terms negative for that to be a solution.
is a solution to the pde.
I.e if your problem read
would be a solution.