Linq to sql datetime issue

The simplest approach is to fetch just the data you want in the part that's translated into SQL, and then do the rest locally:

public List<Class_CourseStartdate> GetClosestStartDate(DateTime coeStartdate)
{
    return _er.TimeTables
              .Where(tt => tt.Startdate == coeStartdate)
              .Select(tt => new { tt.Startdate, tt.TimeTableID })
              .AsEnumerable() // Do the rest locally
              .Select(tt => new Class_CourseStartdate 
              { 
                   CStartdate = tt.Startdate.Value.ToString() ,
                   TimeTableID = tt.TimeTableID 
              }
              .ToList();
}

Note that I've stopped using a query expression here, as it becomes ugly when you need to then call methods on it - and all you're doing is a simple Where and Select. Query expressions are most useful when they involve transparent identifiers, e.g. in joins.

Next up, I would notice that we're already checking that tt.Startdate is equal to the date we've passed in - so why bother fetching it?

public List<Class_CourseStartdate> GetClosestStartDate(DateTime coeStartdate)
{
    return _er.TimeTables
              .Where(tt => tt.Startdate == coeStartdate)
              .Select(tt => tt.TimeTableID)
              .AsEnumerable() // Do the rest locally
              .Select(tid => new Class_CourseStartdate 
              { 
                   CStartdate = coeStartDate.ToString() ,
                   TimeTableID = tid
              }
              .ToList();
}

Or possibly just change the method to return timetable IDs, given that that's really what it's doing. I'd change the method name too:

public List<int> GetTimeTableIdsForStartDate(DateTime coeStartdate)
{
    return _er.TimeTables
              .Where(tt => tt.Startdate == coeStartdate)
              .Select(tt => tt.TimeTableID)
              .ToList();
}

Even if you do stick with Class_CourseStartdate I would change the CStartdate property to be just DateTime rather than string - avoid using strings in your models unless you really, really have to.

Additionally, your naming currently feels like it's all over the place - Startdate should be StartDate everywhere, you should drop the C and Class_ prefixes, etc.

Try this one:

return (from TT in _er.TimeTables
        where TT.Startdate == CoeStartdate
        select new  
        {
            CStartdate = TT.Startdate.Value.ToString() ,
            TimeTableID = TT.TimeTableID 
        }
        ).AsEnumerable()
         .Select(x=> new Class_CourseStartdate
         {
             CStartdate = x.CStartdate.Value.ToString(),
             TimeTableID = x.TimeTableID
         }).ToLIst();

This way you fecth the data from the database and then you apply the formatting you want.

Most functions that are not in defined in LINQ namespace can not be used inside LINQ expressions for IQueryabe (ToString being one of those). The solution is to just copy the value as it is and perform the conversion afterwards, or replace the LINQ with a for loop to iterate through the values, in which case you can use your ToString correctly.

First of all, you have a possible exception there: TT.StartDate may be null and when calling Value you'll get an exception.

Second of all, why is CStartDate a string? You won't be able to perform DateTime specific operations with it and you may encounter problems when converting a string back to DateTime if the conversion is made on a thread with different culture.

As a quick fix, you can try adding a second projection to your query, after loading data in memory:

return _er.TimeTables
    .Where(tt => tt.Startdate == CoeStartdate)
    .Select(tt => new { StartDate = tt.StartDate, Id = tt.TimeTableID})
    .ToList()
    .Select(tt => new Class_CourseStartDate
    {
        // Avoid the exception by checking if property has value
        CStartDate = tt.StartDate.HasValue ? tt.StartDate.Value.ToString() : String.Empty,
        TimeTableId = tt.Id
    })
    .ToList();