Provide examples and evidence of Marxism, existentialism, feminism, and Freudian pscychoanalysis in the Oedipus plays.
In Antigone, Sophocles introduces us to the first great feminist hero, Antigone. She is outspoken against a man, the King. In terms of psychoanalysis, Antigone follows the gods laws (superego) and not the need for revenge (id), thereby exposing Creon's self-serving behavior. In terms of Marxist theory, Antigone is a hero to the proles because refuses to marry the prince and live a life of luxury. Not only that, but she chooses to defy martial law and bury her brother, knowing that she would receive the death penalty. Her existential predicament, though, is whether or not to commit suicide. Rather than have her fate determined for her, she determines her own.
In Oedipus, we have the proletariat suffering from the sins of the bourgeoisie (Marxism): the plague upon the children is brought upon them by the incest and regicide of the Royal House of Thebes. Unlike Antigone, we have Queen Jocasta who follows the patriarchal order to protect her status: she is in denial of the truth and freedom. She, therefore, goes against the ideals of feminism and Marxism. However, Joacasta, is in the same existential predicament, for she commits suicide, not to escape public execution, but because of shame. She, therefore, is no existential hero either. Oedipus is the existential hero: he refuses to kill himself; instead, he punishes himself and suffers the responsibility of knowing the truth.
One of Marx's profoundest insights concerned the way in which inequalities among different classes led to strife; he also claimed that this strife would be the undoing of the ruling classes. One might say that the character of Oedipus illustrates this; he is desperate for power and is willing to subjugate anyone he needs to in order to get it. However, this is very much part of his tragic flaw; perhaps if he weren't so keen to rule, the tragic events would never have unfolded.
In terms of feminism, we might argue that the tragic events could also have been avoided had Jocasta, Oedipus' mother/wife, been allowed a say at any point. Her husband decided unilaterally to send the baby away in order to avoid the prophecy. Oedipus (unknowingly) killed his father and married Jocasta; Jocasta had no say in this matter either.
In term of psychoanalysis, the Oedipus plays led Sigmund Freud to name a complex after him : the Oedipus complex. Freud believed that humans were controlled by drives and desires of which they may not even be aware. One of these drives is the Oedipus complex, the desire of a son for his mother which is usually coupled with a hatred for the father. The central idea is competition with the same-gender parent for the affection of the opposite-gender parent. We can read Oedipus Rex in light of Freud as much as we must read Freud in light of Oedipus Rex.
Though the previous answer covers many factors and schools of thought quite well, there is one more very important thing to know about the relationship between psychoanalysis and Oedipus the King. In the process of originating that field of study and school of criticism, Sigmund Freud used the story of Oedipus to characterize a fundamental concept in psychoanalysis. He argued that all men have a subconscious desire to kill their fathers and marry their mothers. Since Oedipus does exactly that, Freud named this subconscious longing the "Oedipus complex." He coined the term in his work The Interpretation of Dreams. Like most literature, the Oedipus plays certainly benefit from psychoanalytic interpretation, but they are fairly unique in that their central story also serves as the naming example of a key idea in the early days of psychoanalytic thought.