As chapter two begins, Percy's main challenge appears to be the disintegration of his reality. He begins to question his sanity because no one else seems to know who Mrs. Dodds is.
Meanwhile, Mrs. Dodds seems to have been replaced by a Mrs. Kerr, whom Percy has never seen before. To make matters worse, Grover is not able to hide the fact that he remembers Mrs. Dodds and her attack on Percy in chapter one. Percy realizes that everyone is lying to him. At night, he has nightmares about Mrs. Dodds sporting "talons and leathery wings."
Because of his stress, Percy's grades suffer. Essentially, Percy's reality has been disrupted by supernatural events beyond his control. He feels alone, unsure, and frightened. Above all, he misses his mother but knows that he must try to finish out his term at school.
Later, however, Mr. Brunner tells Percy that he must leave Yancy. This adds to Percy's distress. His only comfort is that Grover is taking the same bus into Manhattan. The chapter ends with Percy facing an unknown threat from three old ladies who are knitting a giant pair of socks.
So, Percy's main challenges in chapter two arise from the supernatural events that have contributed to the disintegration of his reality.