Yes DayQuil does work for colds.
Whenever I have a cold I take DayQuil in the daytime to help ease the cold symptoms and it does help.
I then also take NyQuil at night for a cold to help me fall asleep with the cold and it does work.
The DayQuil and NyQuil does not cure the cold but it does help me feel much better and faster and I seem to get over the cold virus much faster taking NyQuil and DayQuil.
Within a week to 14 days most colds are over with and there's nothing really that a doctor can do but you can get plenty of rest, stay hydrated, eat foods with Vitamin C and Vitamin D, Drink orange juice etc and you should recover fully within 7 to 14 days.
But I do recommend taking DayQuil in the day and NyQuil at night to help you recover from the cold virus much sooner.