Connecting SQL SERVER with Initial catalog in C#
You can also supply your database initially by initial catalog option. 
private void btnCheckConnection_Click(object sender, EventArgs e)
        {
            SqlConnection conn = new SqlConnection();
            conn.ConnectionString = "data source=my-PC;integrated security=SSPI;initial catalog=ForgetCode";
            try
            {
                conn.Open();
                MessageBox.Show("Connection was successful");
            }
            catch
            {
                MessageBox.Show("Failed to connect to data source");
            }
            finally
            {
                conn.Close();
            }
        }


Here, ForgetCode is set as default database.