This expression means 'would prefer to', and is followed by the infinitive without to. We often use the contraction 'd rather.
'How about a drink?' 'I'd rather have something to eat'
would rather:past tense with present or future meaning
We can use would rather to say that one person would prefer another or others to do something. We use a special structure with a past tense.
would rather + subject + past tense
I'd rather you went home now.
Tomorrow's difficult. I'd rather you came next weekend.
My wife would rather we didn't see each other any more.
'Shall I open the window?' 'I'd rather you didn't.'
A present tense is sometimes used in this structure (e.g. I'd rather you go home now), but this is unusual.
So, in your example, I would say 'I'd rather you stayed at home.'
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