It couldn’t be more evident that many Christians assume Jesus is the Son of God. But in many cases, Christians aren’t sure about the biblical background of the title “Son of God.” What Christians tend to forget is that when Jewish people thought of the Davidic King as the Son of God, it had very little to do with thinking the King was the Second person of the Trinity. Even though divine sonship appears in the Hebrew Bible with regards to persons or people groups such as angels (Gen 6:2; Job 1:6; Dan 3:25), and Israel (Ex. 4:22-23; Hos 11;1; Mal. 2:10), the category that has special importance to the Son of God issue is the king. When the divine sonship is used in the context of the relationship between Israel and the king (2 Sam. 7:14; Ps. 2:7;89:26-27), the sonship theme emphasizes that the king is elected to a specific task. Furthermore, there is also a special intimacy between God and the king.
The existence of Israel is directly related to God’s covenant with Israel and Israel’s relationship to God as the King. The Davidic covenant established David as the king over all of Israel. Under David’s rule, there was the defeat of Israel’s enemies, the Philistines. David also captured Jerusalem and established his capital there (2 Sam. 1-6). While God promised that Israel would have an earthly king (Gen. 17: 6; 49:6; Deut.17: 14-15), he also promised David that one of his descendants would rule on his throne forever (2 Sam.7:12-17; 1 Chr.17:7-15). In other words, David’s line would eventually culminate in the birth of a specific person who will guarantee David’s dynasty, kingdom, and throne forever. Royal messianism is seen in the Psalms.
For example, in Psalm 2 which is a coronation hymn, (similar to 2 Kings 11:12) is the moment of the king’s crowning. God tells the person to whom he is speaking that He is turning over the dominion and the authority of the entire world to Him (v 8). While David did have conquest of all the nations at that time, (Edom, Moab, Ammon, Philistia, Amalek, which is described as the conquest “of all the nations” 1 Chron. 14:17; 18:11) in Psalm 2, one day God will subjugate all the nations to the rule of the Davidic throne. In Psalm 89, the Davidic King will be elevated over the rivers and seas (v.24- 25) and is the most exalted ruler on earth (v. 27). He also will be the “firstborn” and enjoy the highest rank among all earthly kings. Furthermore, David’s throne continues his dynasty from one generation to the next for perpetuity (vv.28-29). In Psalm 110, the Davidic King is invited to sit at the royal throne at God’s “right hand” (vs.1) and his called “lord” (vs.1) and called a “priest” after the pattern of Melchizedek.
As Israel went into the Babylonian captivity, the prophet Hosea says that Israel will be without a Davidic king for many days (Hosea 3:4).However, in the last days, God kept his promise of the Davidic covenant by rebuilding Israel which includes the re-establishment of the Davidic kingdom (Isa.11:1–2; Hosea 3:5; Amos 9:11–12). The Davidic King will be born in Bethlehem (Mic. 5:2) and would be unlike any past Davidic king (Is.7:14-17; 9:6-7;11:1-10), even though he is not spoken of specifically as “The Messiah.” Ezekiel also spoke of a new David who would be a shepherd as well as a “prince” and a “king” to Israel (Ezek.34:23-24; 37:24-25). There are other texts that speak of the Davidic King as the “Branch” who will reign and rebuild the temple and be a king-priest on His throne (Zech. 3:8; 6:12–15; Jer. 33:1–8, 21–22).
One of the most valuable resources that speak to the Messianic expectation of the time of Jesus is found in The Psalms of Solomon. The Psalms of Solomon is a group of eighteen psalms that are part of the Pseudepigrapha which is written 200 BC to 200 A.D. Even though these works are not part of the Protestant Canon, they are dated just before or around the time of Jesus.
Therefore, they help provide the historian with valuable information about the messianic expectations at the time of Jesus. In it, there are two passages about a righteous, ruling Messiah:
Taught by God, the Messiah will be a righteous king over the gentile nations. There will be no unrighteousness among them in his days, for all shall be holy and their king shall be the Lord Messiah. He will not rely on horse and rider and bow, nor will he collect gold and silver for war. Nor will he build up hope in a multitude for a day of war. The Lord himself is his king, the hope of the one who has a strong hope in G-d. He shall be compassionate to all the nations, who reverently stand before him. He will strike the earth with the word of his mouth forever; he will bless the Lord’s people with wisdom and happiness. And he himself will be free from sin, in order to rule a great people. He will expose officials and drive out sinners by the strength of his word.” (Psalms of Solomon 17.32-36).
John Collins, who is a specialist on this topic says the following about the Davidic Messiah
This concept of the Davidic Messiah as the warrior king who would destroy the enemies of Israel and institute an era of unending peace constitutes the common core of Jewish messianism around the turn of the era. There was a dominant notion of a Davidic Messiah, as the king who would restore the kingdom of Israel, which was part of the common Judaism around the turn of the era.
Even though Luke calls Jesus the “Son of the Most High,” a similar theme was written about in the Qumran literature which predates the New Testament: He will be called the Son of God, and they will call him the son of the Most High…His kingdom will be an eternal kingdom…The earth will be in truth and all will make peace. The sword will cease in the earth, and all the cities will pay him homage. He is a great god among the gods… His kingdom will be an eternal kingdom (4QAramaic Apocalypse (4Q246), col. II:
Collins says the following about 4Q246:
The notion of a messiah who was in some sense divine had its roots in Judaism, in the interpretation of such passages as Psalm 2 and Daniel 7 in an apocalyptic context. This is not to deny the great difference between a text like 4Q246 and the later Christian understanding of the divinity of Christ. But the notion that the messiah was Son of God in a special sense was rooted in Judaism, and so there was continuity between Judaism and Christianity in this respect, even though Christian belief eventually diverged quite radically from its Jewish sources. Collins goes on to concede that even if the dominant Messianic expectation was mostly centered around a Davidic warrior, there is hardly any evidence in the Gospels that accords with the Jewish expectation of a militant messiah. 
Typology and the Davidic King
How do Christians make the leap to Jesus being not only the Davidic King, but divine as well?
There are several ways to answer this. But one way to answer this question is to discuss what is called typology. Some of the features of typology are the following: 1.The prophets did not so much make singular predictions but gave themes or patterns and that these themes have several manifestations or fulfillments in the course of human history. 2.The type and the antitype have a natural correspondence or resemblance. The initial one is called the type (e.g., person, thing, event) and the fulfillment is designated the antitype. 3.The type has historical reality. 4.The type is a prefiguring or foreshadowing of the antitype. It is predictive/prophetic; it looks ahead and points to the antitype. 
Keeping these principles in mind, let’s look at Romans 1:1-5:
Paul, a bond-servant of Christ Jesus, called as an apostle, set apart for the gospel of God, which He promised beforehand through His prophets in the holy Scriptures, concerning His Son, who was born of a descendant of David according to the flesh, who was declared the Son of God with power by the resurrection from the dead, according to the Spirit of holiness, Jesus Christ our Lord, through whom we have received grace and apostleship to bring about the obedience of faith among all the Gentiles for His name’s sake, among whom you also are the called of Jesus Christ; to all who are beloved of God in Rome, called as saints: Grace to you and peace from God our Father and the Lord Jesus Christ.” In this text, Paul says through the resurrection, Jesus is installed (by God) as the Son of God (Rom. 1:4).
Paul is not saying Jesus is being appointed as The Son of God is a change in Jesus’ essence. As N.T. Wright says, “The appointment is not in terms of his nature but in terms of his work as the Davidic king—the messianic age has dawned. Jesus is the anti-type of the previous “sons” in the Old Testament (Adam, David, Israel).” The New Testament authors unanimously declare Jesus as the one who is from the “seed of David,” sent by God to restore God’s kingship over mankind (Matt. 1:1; Acts 13:23; Rom. 1:3,4; 2 Tim:2:8; Rev. 22:16).
As seen in 2 Samuel 7:12-17, the immediate prophecy is partially fulfilled in David’s son Solomon. However, as already said, the word “forever” shows there are future descendants to come. God promised David that his “seed” would establish the kingdom. There were two ways for this prophecy to come to pass. Either God could continually raise up a new heir or he could have someone come who would never die. Does this sound like the need for a resurrection? As Murray Harris says, There is a loose parallel in the case of a royal family where a child is ‘born’ a king but subsequently ‘becomes’ king at his coronation. From this standpoint, the resurrection was the coronation or installation of Jesus as the Son of God.” 
Also, In Psalm 2:11 and Psalm 100:2, the rulers and the people are supposed to worship and serve the Lord, while in Psalm 18:44 and Psalm 72:11 it says it is the Davidic king whom they must worship and serve. This theme makes perfect sense in the New Testament passage, John 5:22-23, “Moreover, the Father judges no one, but has entrusted all judgment to the Son, that all may honor the Son just as they honor the Father. He who does not honor the Son does not honor the Father, who sent him.”
Sources:  Herbert W. Bateman IV, Darrell L. Bock, and Gordon H. Johnston, Jesus the Messiah: Tracing The Promises, Expectations, And Coming of Israel’s King ( Grand Rapids: Kregel Academic, 2012), 80.  Ibid, 97.  John J. Collins, The Scepter and the Star: The Messiah of the Dead Sea Scrolls and Other Ancient Literature (New Haven, Conn.: Yale University Press, 2007), 68, 209. Ibid, pp. 168-169.  Ibid, pp.13, 204.  H.Wayne House and Randall Price, Charts of Bible Prophecy (Grand Rapids: Zondervan, 2003), 35.  Harris, M. Raised Immortal: Resurrection and Immortality in the New Testament. Grand Rapids, MI: Wm. B. Eerdmans Publishing Co. 1983, 74-75.