After this there was a feast of the Jews, and Jesus went up to Jerusalem.
John 5 places the context as during a feast of the Jews, but the particular feast is not specified. Some commentators suggest it is inconsequential, but simply intended to explain why Jesus is now found in Jerusalem. Some, however, suggest that the identification of the feast has relevance in regard to determining the duration of Jesus’ public ministry. For instance the Jamieson-Fausset-Brown Bible Commentary has this to say:
1. a feast of the Jews—What feast? No question has more divided the Harmonists of the Gospels, and the duration of our Lord’s ministry may be said to hinge on it. For if, as the majority have thought (until of late years) it was a Passover, His ministry lasted three and a half years; if not, probably a year less. Those who are dissatisfied with the Passover-view all differ among themselves what other feast it was, and some of the most acute think there are no grounds for deciding. In our judgment the evidence is in favor of its being a Passover, but the reasons cannot be stated here.
To help clarify why the year variation in the public ministry of Jesus is relevant to identifying this feast as Passover is explained more from Meyer’s NT Commentary:
If this feast itself is taken to be the Passover, we are obliged, with the most glaring arbitrariness, to put a spatium vacuum of a year between it and the Passover of John 6:4, of which, however, John (John 6:1-4) has not given the slightest hint. On the contrary, he lets his narrative present the most uninterrupted sequence. Hengstenberg judges, indeed, that the gap can appear strange only to those who do not rightly discern the relation in which John stands to the Synoptics. But this is nothing more than the dictum of harmonistic presuppositions.
The Expositor’s Greek Testament succinctly sums up the two predominant and competing views:
It is chiefly between Purim and Passover that opinion is divided, because some feast in spring is supposed to be indicated by John 4:35. Against Passover it is urged that in chap. 6 another Passover is mentioned; but this is by no means decisive, as John elsewhere passes over equally long intervals of time. Lampe, Lightfoot, Grotius, Whitelaw, and Wordsworth argue for Passover: Tischendorf, Meyer, Godet, Farrar, Weiss, and others strongly favour Purim; while Lücke seems to prove that no sure conclusion can be reached. [For a full and fair presentation of opinions and data see Andrew’s Life of our Lord, p. 189 sqq.] The feast, whatever it was, is mentioned here to account for Jesus being again in Jerusalem.