Though Harry knows Hermione was trying to help, he's still angry with her. Ron is furious. Hermione begins avoiding the common room and Harry and Ron do nothing to try to make her come back. Right before term starts, Oliver finds Harry to ask about his progress on the dementors and on ordering a new broom. When he learns that McGonagall confiscated Harry's brand new Firebolt, Oliver isn't at all concerned that Black is out to get Harry and promises to talk to McGonagall.
Harry at least understands why Hermione did what she did, which suggests that he'll be able to forgive her more easily than Ron, who still hasn't fully forgiven Hermione for Crookshanks's constant attempts on Scabbers' life. This indicates that, in terms of maturity, Ron has the most progress to make of the three.
The next day, Hagrid teaches an enjoyable lesson on salamanders that love fire, while Trelawney introduces palmistry and tells Harry his life line is very short. After Defense Against the Dark Arts, Harry asks Lupin if they can start their anti-dementor lessons. Lupin asks Harry to meet him Thursday evening. As Harry and Ron head for dinner, Ron remarks that Lupin still looks unwell and wonders what's wrong with him. They hear Hermione behind them and she says it's obvious what's wrong. Ron won't indulge her and after she walks off, he tells Harry that she's just trying to get them to talk to her.
Note that Ron never considers that Hermione might actually be right—and indeed, she reveals later that she does know what's ailing Lupin. This illustrates how Ron's heightened emotions make him unwilling to listen to or consider any perspectives other than his own, as the most important thing for him right now is making sure Hermione understands how upset he is with her.
On Thursday evening, Harry meets Lupin in the History of Magic classroom. Lupin brings a suitcase containing a boggart. Harry tries to hide his apprehension as Lupin tells him about the Patronus charm, which conjures a positive force that a dementor can feed on. To conjure it, a wizard must think of a happy memory and say, "Expecto Patronum." Harry thinks of his first time on a broomstick and something silvery and wispy comes out of his wand. As Lupin opens the case, Harry tries to think of flying. He shouts the incantation and as he passes out, he hears Lily pleading with Voldemort.
As in his other lessons, Lupin walks Harry through the process in a way that builds on previous knowledge and, most importantly, he chooses to work with a boggart since it'll save Harry from most of the danger posed by real dementors. This shows that he understands how to both work within Dumbledore's rules and create an educational situation that offers Harry the best chance for success.
Harry jerks back awake on the floor, apologizes, and accepts Lupin's proffered Chocolate Frog. As he eats it, he mutters that he heard Lily and Voldemort louder this time, and when Lupin suggests they call it a night, Harry insists he has to learn to fight the dementors. Lupin suggests that Harry choose a stronger memory, so Harry thinks of winning the House Cup last year. This time, Harry hears James telling Lily to take Harry and run. When he comes to, Harry is crying. Lupin admits that he was friends with James and says that the charm is too advanced.
Lupin's suggestion to stop the lessons suggests that hearing from Harry that he hears his parents, old friends of Lupin's, is hard for Lupin as well as Harry. When Harry learns that his father and Lupin were friends, he's able to connect with Lupin on a more personal level and deepen their relationship further.
Harry insists on trying again and settles on the memory of learning he was a wizard. Lupin opens the case, Harry shouts the incantation, and though he still hears Lily screaming, it's quieter. A shadow bursts out of Harry's wand and after a few seconds, Lupin shouts "Riddikulus" and the boggart dementor vanishes. Lupin praises Harry, gives him a huge bar of chocolate, and asks Harry to come again next week. Before Harry leaves, he asks Lupin if he also knew Sirius Black. Lupin looks suspicious, but just says that he didn't know Black as well as he thought he did.
Lupin's suspicion indicates that he might guess Harry knows more about Sirius Black than Harry has let on. When Lupin says he didn't know Black as well as he thought he did, it indicates that when Black betrayed Lily and James, it came as a shock to everyone. Lupin's unwillingness to truly level with Harry, however, indicates that he's not sure if it's his place to give Harry this truth.
After the Quidditch match between Ravenclaw and Slytherin, Oliver increases practice to five days per week. This leaves Harry one night a week to do homework, though he still doesn't look as tired as Hermione does. One evening, Ron quietly asks Harry how Hermione is getting all her work done—it seems impossible that she's making it to all her classes, but she hasn't missed one. Oliver interrupts to say that he's spoken to McGonagall about the Firebolt, and that she took great offense when he asked if Harry could have it back. He suggests Harry order a new broom.
Again, it's a step in the right direction that Harry and Ron recognize that there's something strange going on with Hermione. However, their unwillingness to dig deeper to figure out what's going on suggests that they either trust her enough to tell them the truth, that they fear her snappy reaction if they do ask, or that they're simply not curious or interested enough to investigate further.
McGonagall continues to refuse to give Harry the Firebolt. To make things worse, the dementor lessons with Lupin stall and Harry's Patronus never gets distinct. After four weeks, Lupin tells Harry that for his age, his indistinct Patronus is huge. He pulls out butterbeer as a treat for Harry, assuming Harry has never had it. Harry tries to play along, but Lupin still looks suspicious. After a moment, Harry asks what's under a dementor's hood. Lupin says that nobody really knows, as people only see it when dementors administer the Dementor's Kiss, which sucks the soul out of a victim and leaves them an empty shell. Lupin says that Sirius Black will suffer this fate if the dementors catch him. Harry spits that Black deserves it and wants to tell Lupin about what he heard in the Three Broomsticks, but he doesn't want to admit he snuck into Hogsmeade.
When Harry wants to share what he learned in Hogsmeade with Lupin, it shows that he desperately craves an adult perspective on this information, preferably from an adult he trusts so much. However, his desire to not disappoint Lupin by admitting to sneaking out of school shows that Harry also wants to make sure that Lupin will continue to think highly of him. At this point, Harry believes that the only way to remain in Lupin's good graces is to pretend that he's being a model student and following all instructions.
On his way back upstairs, Harry runs into McGonagall and is surprised when she gives him his Firebolt. She tells him to try to win on Saturday before she walks off. Ron finds Harry in the hallway to ask if he can ride it, and Harry suggests they make up with Hermione. When they get to the portrait hole, they find Neville pleading with Sir Cadogan. Apparently, Neville wrote down the passwords for the week and lost his list. Harry gives Neville the password and as soon as they get into the common room, Gryffindors swarm to look at the Firebolt. After ten minutes, Harry and Ron finally approach Hermione. Ron offers to take the Firebolt upstairs and Harry sits with Hermione.
McGonagall's request that Harry do his best to win shows that she's not entirely impartial when it comes to her students, but she keeps her moments of impartiality to a venue where it's entirely appropriate—Quidditch, where the whole point is for one House to win and the others to lose. This again shows that McGonagall is a far superior teacher to Snape, who plays favorites all the time and not just in appropriate instances.
Harry looks at all of Hermione's books and suggests she drop a few subjects. Hermione looks scandalized and suddenly, they hear a yell from the boys' dormitory. Ron races downstairs with a bed sheet and an angry expression. There's a spot of blood on it, and he throws ginger cat hairs onto Hermione's essay.
Harry's suggestion shows that he cares deeply for Hermione and sees that what she's doing is having negative effects on her mental health. With this, he's able to show that he's growing up and becoming more empathetic.