How to remember the difference between: "Can you try to open" and "Can you try opening"?

Solution 1:

If you try to do something, you make an attempt to do that something.

The use of the word 'try' followed by a to- infinitive suggests that you are not sure whether you will succeed in a future attempt: I am trying to open the window. I will try to open the window.

It suggests you did not succeed in a past attempt: I tried to open the window.

If you try doing something, you actually do it. The use of the word 'try' followed by a gerund suggests that you are want to see if the result of what you do will have some effect: The room is very stuffy; I will try opening the window (to see if that helps).

'Can you ...?' introducing a request to somebody to try to do / try doing something does not change the meanings of the try constructions.

Can you try to open the window? This is request to somebody to make the attempt to open the window. The speaker is not sure whether the person addressed will succeed. It is possible that somebody else has already tried, and failed.

Can you try opening the window? This is a request to somebody to open the window. The speaker hopes that the opening of the window will have some effect, such as letting some fresh air into a stuffy room.

Solution 2:

Try + infinitive = make an effort to do something

Try + gerund = experiment to see if something works or make an effort to do something

The catenation with the to-infinitive has an unambiguous meaning.

The catenation with the ing-form may take either of the given meanings.

Context will often indicate which meaning is intended here:

-It's stuffy in here - I'm feeling rather light-headed.

-Try opening the window.

-Have you tried opening it recently?

If ambiguity remains, the statement should be rephrased:

See if running three miles every morning makes an improvement.