Could you let me know if these two phrasal verbs have the synonymous meanings ?
a) I'd better to go through with the homework.
b) I'd better to get the homework over with.
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They have a similar meaning, but the nuance is different.
go through (with something) (verb phrase): To do something unpleasant or difficult that has already been agreed or promised.
This phrase normally implies that the person doing the task is reluctant to carry it out.
We all knew Stephen was afraid of heights, so when we challenged him to come skydiving with us we never thought he'd go through with it. But he surprised us all.
While you would think this makes sense to use for homework, it doesn't sound natural for something so trivial. The phrase is normally used for more significant events, for which there is some implied moral obligation.
get (something) over with: to finish or reach the end of some unpleasant work, experience, or duty:
In comparison, this means to reluctantly get something done, with more of a focus on completion, than the process of doing it. There may be a sense of responsibility or duty to accomplish some task, but not the same level of obligation as with "get through". You may also "get it over with" in any way that seems appropriate to you, rather than as a fulfillment of a promise.
Sam didn't want to make all those fundraising calls for his campaign. He'd much rather be talking to voters, or the press, or anyone other than wealthy donors. But if he wanted to have the money to win the election, he'd best get them over with.
This phrase is appropriate to use for homework, especially if you don't want to do it.
Note: "Get through (something)" does have a similar meaning as "get (something) over with".
She figured, if she could only find a way to get through Mondays, the rest of the week would be a cinch.
As with "go through" this focuses a bit more on the process of reaching completion, but it lacks the feeling of moral obligation.
I'd better get through this homework, before we start playing video games.
They could in some cases describe the same activity objectively defined, but they have a different emotional meaning.
"Go through with" means to continue and complete some action that you have committed to doing, or been charged with doing. There is an implication that the person is unwilling, but it is not necessarily difficult or unpleasant to do it (it might just be the consequences that are undesirable). It could be a quick or a lengthy activity, but it has a definite end-point. The alternative to going through with it is just not doing it - maybe not starting, or maybe leaving it incomplete.
"Get something over with" means to finish it - not necessarily completing it, or doing it satisfactorily, just getting to the end and stopping. There is an implication that the activity itself is difficult or unpleasant. The alternative might be not doing it at all (if it is a short activity), but it might be just letting it drag on inconclusively.
In the case of the homework, "get it over with" is much more natural, to my ear.