終わる = 宿題が終わる (Homework is done) = 仕事が終わる (Work is over) 終える = 宿題を終える (finish one's homework) = 仕事を終える (finish one's work) 済む = 宿題が済む (Homework is done) = 仕事が済む (Work is over) 済ませる = 宿題を済ませる (finish one's homework) = 仕事を済ませる (finish one's work) 済む can also be used as "solve." = 話し合いで済む問題 (A problem that can be solved (only) by discussions.) 済ませる = 問題を話し合いで済ませる (solve a problem by discussions.) I'm sorry if my translations were wrong or not clear enough!

I am afraid my explanation might not be helpful. 終わる・・・we use this when something end spontaneously. ex)授業が終わる. subject must be something.(not somebody) 終える・・・we use this when somebody end something. ex) 授業を終える。 subject must be something.(not somebody) this also emphasizes you finished it. 済む・・・we often use this with other words. what i first come up with is しなくて済む。 ex)　そうじが済む。 　　　そうじをしなくて済んだ。 I think it's better to use 終わる　instead of 済む.Because in some cases we can not use 済む　instead of 終わる.We don't say like this. ex)国会（こっかい）が済む。 　　２人の関係（関係）が済む。 i hope you enjoy process to improve your language.

終わる is intransitive, as is 済む 終える is transitive There is a transitive variant for 済む, but I'll leave you to find that by yourself. BTW, dozens of Japanese verbs come in transitive-intransitive pairs, so I suggest you take the time to get to grips with it.