if you want to do it only with the LocalDate-class:
LocalDate initial = LocalDate.of(2014, 2, 13);
LocalDate start = LocalDate.of(initial.getYear(), initial.getMonthValue(),1);
// Idea: the last day is the same as the first day of next month minus one day.
LocalDate end = LocalDate.of(initial.getYear(), initial.getMonthValue(), 1).plusMonths(1).minusDays(1);