Final answer:
In response to Patroclus' death, Achilles seeks revenge against Hector, returning to battle with new armor forged by Hephaestus.
Step-by-step explanation:
After finding out about Patroclus' death, Achilles is overcome by grief and rage. He decides to return to the battle, even though he had previously withdrawn due to a dispute over the captive Briseis with Agamemnon. In his fury, Achilles does not simply mourn; he seeks vengeance and is determined to face Hector, whom he holds responsible for Patroclus' demise.
This is the catalyst that brings Achilles back into the fold of combat, leading to one of the most famous duels in Homeric literature and indeed Western literature at large. He also requests his mother, Thetis, to ask Hephaestus to forge new armor for him, signifying his return to war and highlighting the divine intervention often present in Greek mythology.