You can get the weekday from the DateTime using https://api.dart.dev/stable/2.5.1/dart-core/DateTime/weekday.html and add/subtract this number from you date:

void main() {
  final date = DateTime.parse('2019-10-08 15:43:03.887');
  
  print('Date: $date');
  print('Start of week: ${getDate(date.subtract(Duration(days: date.weekday - 1)))}');
  print('End of week: ${getDate(date.add(Duration(days: DateTime.daysPerWeek - date.weekday)))}');
}

DateTime getDate(DateTime d) => DateTime(d.year, d.month, d.day);

UPDATE

Please read and upvote the answer from lrn. He knows a lot more about this stuff than me. :)

Dart DateTimes have a weekday getter which is 1 for Monday and 7 for Sunday. Using that, I would do:

DateTime mostRecentSunday(DateTime date) =>
    DateTime(date.year, date.month, date.day - date.weekday % 7);

to get the most recent Sunday (which is the start of the current week if the week starts on a Sunday), and

DateTime mostRecentMonday(DateTime date) =>
    DateTime(date.year, date.month, date.day - (date.weekday - 1));

for the most recent Monday (which is then the start of the current week if the week starts on a Monday).

You can generalize to

/// The [weekday] may be 0 for Sunday, 1 for Monday, etc. up to 7 for Sunday.
DateTime mostRecentWeekday(DateTime date, int weekday) =>
    DateTime(date.year, date.month, date.day - (date.weekday - weekday) % 7);

If you are going to be using the results as calendar dates, I'd used DateTime.utc as the constructor instead. (Always use UTC for calendar dates, then you can do day-based arithmetic on them safely).

I'd even consider using DateTime.utc in any case because it avoids any potential issues with a daylight saving that starts at midnight (rare, but with time zones, even unlikely things tend to have happened somewhere at some point).

The way you do this will probably depend on what your app's localization is. If you want to treat Sunday as the start of the week, you would do this:

DateTime now = DateTime.now();
int currentDay = now.weekday;
DateTime firstDayOfWeek = now.subtract(Duration(days: currentDay));

If you are treating Monday as the start, then do this:

DateTime now = DateTime.now();
int currentDay = now.weekday;
DateTime firstDayOfWeek = now.subtract(Duration(days: currentDay - 1));

MaterialLocalizations

If you're using Dart in Flutter, you can use the MaterialLocalizations class to get the index of the first day of the week (0 = Sunday, 6 = Saturday). This should help you decide which method from above that you should use.

The example given for the firstDayOfWeekIndex property is a good reference:

var localizations = MaterialLocalizations.of(context);
// The name of the first day of week for the current locale.
var firstDayOfWeek = localizations.narrowWeekdays[localizations.firstDayOfWeekIndex];

The MaterialLocalizations class comes with a ton of built in methods to format and display DateTimes.

dateTime: 2021-01-26 11:21:05.429320

• formatFullDate: Tuesday, January 26, 2021
• formatCompactDate: 01/26/2021
• formatMediumDate: Tue, Jan 26
• formatShortDate: Jan 26, 2021
• formatShortMonthDay: Jan 26
• formatMonthYear: January 2021
• formatYear: 2021

If none of these fit your needs, you can also use the DateFormat class to specify how the DateTime should be displayed. It's important to note that the DateFormat class indexes days differently where 1 = Monday and 7 = Sunday.

DateFormat.yMMMd().format(new DateTime.now()) // Jan 26, 2021
DateFormat(DateFormat.ABBR_MONTH_DAY).format(now) // Jan 26
DateFormat(DateFormat.WEEKDAY).format(now) // Tuesday

FIRST DAY OF THE WEEK

DateTime findFirstDateOfTheWeek(DateTime dateTime) {
    return dateTime.subtract(Duration(days: dateTime.weekday - 1));
}

LAST DAY OF THE WEEK

DateTime findLastDateOfTheWeek(DateTime dateTime) {
    return dateTime
        .add(Duration(days: DateTime.daysPerWeek - dateTime.weekday));
}

LAST DAY OF THE MONTH

DateTime findLastDateOfTheMonth(DateTime dateTime) {
    return DateTime(dateTime.year, dateTime.month + 1, 0);
}

FIRST DAY OF THE MONTH

DateTime findFirstDateOfTheMonth(DateTime dateTime) {
    return DateTime(dateTime.year, dateTime.month, 1);
}

LAST DAY OF THE YEAR

DateTime findLastDateOfTheYear(DateTime dateTime) {
    return DateTime(dateTime.year, 12, 31);
}

FIRST DAY OF THE YEAR

DateTime findFirstDateOfTheYear(DateTime dateTime) {
    return DateTime(dateTime.year, 1, 1); }

Dart/Flutter – How to find the first date and the last date of a week

1. Find the first date of the week

/// Find the first date of the week which contains the provided date.
DateTime findFirstDateOfTheWeek(DateTime dateTime) {
  return dateTime.subtract(Duration(days: dateTime.weekday - 1));
}

2. Find the last date of the week

/// Find last date of the week which contains provided date.
DateTime findLastDateOfTheWeek(DateTime dateTime) {
  return dateTime.add(Duration(days: DateTime.daysPerWeek - dateTime.weekday));
}

Testing

void main() {
  // Find first date and last date of THIS WEEK
  DateTime today = DateTime.now();
  print(findFirstDateOfTheWeek(today));
  print(findLastDateOfTheWeek(today));

  // Find first date and last date of any provided date
  DateTime date = DateTime.parse('2020-11-24');
  print(findFirstDateOfTheWeek(date));
  print(findLastDateOfTheWeek(date));
}

// Output
2020-11-23 06:54:42.865446
2020-11-29 06:54:42.865446
2020-11-23 00:00:00.000
2020-11-29 00:00:00.000

None of the above worked for me but @JoeMuller gave an interesting piece of info on the material localizations, which led me to find out that for what i want, sunday in date.weekday should be 0 took a while to figure it out many thanks to joe and @julemand101 I have this for UK calendar starting from sunday and ending on saturday

void main() {
   final date = DateTime.parse('2021-08-01');

   print('Date: $date');
   final weekDay =  date.weekday == 7 ? 0 : date.weekday;
   print('Start of week: ${getDate(date.subtract(Duration(days: weekDay)))}');
   print('End of week: ${getDate(date.add(Duration(days: DateTime.daysPerWeek - weekDay - 1)))}');
}

DateTime getDate(DateTime d) => DateTime(d.year, d.month, d.day);

Below is the code which should work fine in most cases.

We can get start and end of the week like this:-

To get the first day of the week for a date use the below function. Like Sunday (Also known as 7 in WeekDay).

Read lines of comments in code for code clarifications.

DateTime getFirstDayOfWeek({required DateTime currentDateTime}) {
  // Converting date provided to UTC
  // So that all things like DST don't affect subtraction and addition on date
  DateTime dateTimeInUTC = DateTime.utc(
      currentDateTime.year, currentDateTime.month, currentDateTime.day);

  // Getting weekday for the date
  // For reference Sunday weekday is 7 and Friday weekday is 5
  int currentWeekDayInUTC = dateTimeInUTC.weekday;

  // Getting Date for nearest Sunday from the provided date
  // By going back a number of weekdays from the current date to reach Sunday
  DateTime firstDayOfWeekInUTC;
  // If current date is not Sunday subtract days to reach Sunday
  if (currentWeekDayInUTC != DateTime.sunday) {
    firstDayOfWeekInUTC =
        dateTimeInUTC.subtract(Duration(days: currentWeekDayInUTC));
  }
  // If current date is Sunday use it as the first day of week
  else {
    firstDayOfWeekInUTC = dateTimeInUTC;
  }

  // Converting back the date for Sunday from UTC type to Local
  // You can also use UTC type depending on your use case
  DateTime firstDayOfWeekInLocal = DateTime(firstDayOfWeekInUTC.year,
      firstDayOfWeekInUTC.month, firstDayOfWeekInUTC.day);

  if (currentDateTime.isUtc) {
    return firstDayOfWeekInUTC;
  } else {
    return firstDayOfWeekInLocal;
  }
}

To get the last day of the week for a date use the below function. Like Saturday (Also known as 6 in WeekDay).

Read lines of comments in code for code clarifications.

DateTime getLastDayOfWeek({required DateTime currentDateTime}) {
  // Converting date provided to UTC
  // So that all things like DST don't affect subtraction and addition on date
  DateTime dateTimeInUTC = DateTime.utc(
      currentDateTime.year, currentDateTime.month, currentDateTime.day);

  // Getting weekday for the date
  // For reference Sunday weekday is 7 and Friday weekday is 5
  int currentWeekDayInUTC = dateTimeInUTC.weekday;

  // Getting Date for nearest Saturday from the provided date
  // By going forward a number of weekdays from the current date to reach Saturday
  DateTime lastDayOfWeekInUTC;
  // If current date is not Sunday add days enough to reach Saturday
  if (currentWeekDayInUTC != DateTime.sunday) {
    lastDayOfWeekInUTC = dateTimeInUTC
        .add(Duration(days: DateTime.saturday - currentWeekDayInUTC));
  }
  // If current date is Sunday add days UpTo saturday
  else {
    lastDayOfWeekInUTC = dateTimeInUTC.add(Duration(days: DateTime.saturday));
  }

  // Converting back the date for Sunday from UTC type to Local
  // You can also use UTC type depending on your use case
  DateTime lastDayOfWeekInLocal = DateTime(lastDayOfWeekInUTC.year,
      lastDayOfWeekInUTC.month, lastDayOfWeekInUTC.day);

  if (currentDateTime.isUtc) {
    return lastDayOfWeekInUTC;
  } else {
    return lastDayOfWeekInLocal;
  }
}