"Methods to do" vs "Methods for doing"

Solution 1:

At least from the predominant usage point, they do not mean the same thing.

There may be any number of methods 'to do' something but usually just one or a few methods 'for doing' something.

"I want to find a method to solve this equation." "This is the method for solving this equation."

Note the difference in article use: a method to; the method for.

Solution 2:

I would go with "methods of doing something". COCA and BNC agree, giving more hits for "methods of doing" than for "methods for doing" and "methods to do" combined.

                      COCA    BNC

methods of doing       21      21
methods for doing      10       1
methods to do           8       0