英语词源字典 - 在线词源网站

buck (v.)[buck 词源字典]: 1848, apparently with a sense of "jump like a buck," from buck (n.1). Related: Bucked; bucking. Buck up "cheer up" is from 1844.[buck etymology, buck origin, 英语词源]
cheer (v.): late 14c., "to cheer up, humor, console;" c. 1400 as "entertain with food or drink," from cheer (n.). Related: Cheered; cheering. Sense of "to encourage by words or deeds" is early 15c. Which had focused to "salute with shouts of applause" by late 18c. Cheer up (intransitive) first attested 1670s.
comfort (v.): late 13c., conforten "to cheer up, console," from Old French conforter "to comfort, to solace; to help, strengthen," from Late Latin confortare "to strengthen much" (used in Vulgate), from Latin com-, intensive prefix (see com-), + fortis "strong" (see fort). Change of -n- to -m- began in English 14c. Related: Comforted; comforting.
hearten (v.): 1520s, "put heart into" (transitive), from heart (n.) in the figurative sense + -en (1). Intransitive sense "to cheer up" is from 1708. Related: Heartened; heartening. Earlier verb was simply heart (Old English).
relieve (v.): late 14c., "alleviate (pain, etc.), mitigate; afford comfort; allow respite; diminish the pressure of," also "give alms to, provide for;" also figuratively, "take heart, cheer up;" from Old French relever "to raise, relieve" (11c.) and directly from Latin relevare "to raise, alleviate, lift up, free from a burden," from re-, intensive prefix (see re-), + levare "to lift up, lighten," from levis "not heavy" (see lever).

The notion is "to raise (someone) out of trouble." From c. 1400 as "advance to the rescue in battle;" also "return from battle; recall (troops)." Meaning "release from duty" is from early 15c. Related: relieved; relieving.