2003 Corolla LE. No engine codes or unusual sounds. Blows cold air for 30 minutes then heat comes on for 2 minutes, then another 10 minutes of cold air, then heat finally comes on and stays on.
What is your temp needle at when it does that? If the car doesn't come up to normal operating temperature even after 30 mins (or if the temp needle goes down at idle), then you likely have a thermostat that is stuck in the open position.
If the operating temperature is normal, it is an issue with the heater core or temperature knob switcher mechanism/system.
IMHO...most likely air in system. Have you recently changed hoses, radiator, thermostat, water pump? With car cold (morning) open radiator cap... Radiator full of coolant or not. How about overflow bottle, up to cold level, or empty?