I can't get a Sql Server localdb connection to work on a computer that does not have SqlServer Express installed

The problem was related to having Sql Server Express LocalDB 2014 installed instead of 2012. With that version MS has changed the connection string requirements. Instead of Data Source=(LocalDB)\V11.0, the connection string is Data Source=(LocalDB)\MSSQLLocalDB. After changing my connection string the program is running correctly on a computer that only has the LocalDB 2014 installed. Here is a link to an article about it: https://connect.microsoft.com/SQLServer/feedback/details/845278/sql-server-2014-express-localdb-does-not-create-automatic-instance-v12-0

also

http://msdn.microsoft.com/en-us/library/hh510202(v=sql.120).aspx

Using "Data Source=(LocalDB)\MSSQLLocalDB" also not worked form me. I had to access databasseusing "Data Source=(LocalDB)\V12.0" and for working that access to work I needed to run this command first "sqllocaldb create "v12.0". More details on this link https://dyball.wordpress.com/2014/04/28/sql-2014-localdb-error-cannot-connect-to-locaidbv12-o/

You'll want to make sure that you've installed .NET Framework 4.0 and, equally as importantly, the .NET Framework 4.0.2 update (KB #2544514).

Once your system is up to date, you can download the SqlLocalDb installer from:

http://www.microsoft.com/en-us/download/details.aspx?id=29062