As you said, SelectedItem is read-only. But, SelectedValue is not.
If you have two items in your dropdownlist with string values of "True" and "False", then you just set the SelectedValue property as in
SolvedDropDownList.SelectedValue = solved;
But that does depend on what solved is and what is contained in the items in your dropdownlist.
If you did this:
SolvedDropDownList.Items.Add(true);
SolvedDropDownList.Items.Add(false);
it would add two items with the string representations of true and false.
Then, if solved was a boolean you should be fine.