Yes you can wash bed sheets in the washing machine and bed sheets wash best in the washing machine on the cold water cycle.
However if you have the flu or a cold then it would be best to sanitize the bed sheets by washing them in the warm hot water cycle.
I wash my bed sheets once per week in the washing machine using regular laundry detergent and some apple cider vinegar to help kill any bacteria that may be on the bed sheets.
I do that especially during cold and flu season to get rid of any germs from the bed sheets.
But you can use regular laundry detergent even during cold and flu season washing your bed sheets but the apple cider vinegar helps to kill off bacteria.