i think it's easier to get to know it separately because of its typical besides interesting structure: these two words are made of [a noun+a verb], in your case, 气->noun 馁->verb 灰->verb 心->noun 气：the energy 馁：weaken 气馁：the energy has been weakened / your vital energy is going down which generally implies you have been beaten once or many times by sth/sb after having tried damn hard so that you are getting 气馁。 灰：discourage, the grey color usually represents a sad one, so discourage people. 心：the heart 灰心：the color of your heart right now is turning grey－> you have been discouraged, that you are not getting confidence to do it over again, you are 灰心 towards sth. 气馁 is giving emphasize on the energy, [physically] while 灰心 is caring more about your feeling, [mentally].

气馁：指失去信心与勇气,提不起精神再做任何事。short of spirit to do something. 灰心：因失败而丧失勇气或信心。lose heart, and heart / confidence turn grey. 总的来说，两者相似。灰心较常用在口头语。

good question. u should try to use www.baidu.com because these questions are hard even for chinese. 灰心：u do something but fail, so u lose ur confidence. 气馁：u do something but fail, so u dont want to do anything, u just drink alcohol, sleep day and night and so on. for many cases they are nearly the same but still, they are different.

气馁 -- 说的是气. 气,在中国是指人体内的一种无形的能量. 灰心 -- 说的是心. 心,在中国是指人的一种比气更抽象的能量.

气馁情感更强，书面语。 灰心情感较弱，口头语。 意思都是差不多的。