Surprisingly many problems are similar, so in the end you can often apply the methods from one domain to another. If you use your study time well you will remember the general principles even though you might forget the exact implementations and most of the details.
Most importantly, you have a much greater chance of knowing when a solution exists and where to look for it, and more importantly know when no solutions are known.
E.g., the computer scientist recognizes the travelling salesman problem even when it doesn't involves salesmen, travelling or cities. She knows that it's NP-hard so she doesn't spend a few months trying to find an efficient exact solution for her millions of nodes, but she knows where to look to find "good enough" solutions for the problems she is trying to solve.
The autodidakt might not even recognise his problem as a TSP and thus doesn't even know that her problem is hard and which books she needs to start reading.
Most importantly, you have a much greater chance of knowing when a solution exists and where to look for it, and more importantly know when no solutions are known.
E.g., the computer scientist recognizes the travelling salesman problem even when it doesn't involves salesmen, travelling or cities. She knows that it's NP-hard so she doesn't spend a few months trying to find an efficient exact solution for her millions of nodes, but she knows where to look to find "good enough" solutions for the problems she is trying to solve.
The autodidakt might not even recognise his problem as a TSP and thus doesn't even know that her problem is hard and which books she needs to start reading.