Chosen Solution
When I start a cold engine of Toyota Corolla 2004, it gives jerks for first 5 to 10 minutes, and then it gets normal.
Could be in need of a good service, it is a 13 year old car so will produce some foibles. New Air Filter, Oil Filter, Spark plugs and a change of oil should breathe new life into the car. If you’re trying to save pennies then I would suggest just taking out the spark plugs, if they have sooty deposits on them try giving them a clean and putting them back, this might give it a better spark and reduce the jumpiness. NOTE: When removing spark plugs and leads, ensure you note which is which as they order they fire is important, or simply just do one at a time.