I am an American Native English speaker. We would use "on my own" or "by myself". To say "I study by my own." is grammatically correct, but never used. They all mean the same thing. By my own is used with a noun at times. Examples: By my own effort, I made a good grade. By my own recipe, I made the best cake. These are still rarely heard (at least as far as in American English). I would stick to "on my own" and "by myself". The English version of one of the songs in the musical Les Misérables is called "On my own". =]

In the West is being in the western region of an area. I am in the Western United States. I am in West Philadelphia. I am in the West. On the west is used in rare circumstances such as the following: On the west coast, I go surfing. On the west side of town, there are many cars. On the western frontier, people made their new lives. Famous Book title: All Quiet on the Western Front You can say "I am in the west." You would never just say "I am on the west." There is not grammatical rule for using them in phrases. It is just memorizing. Hope that helps.

My desenvoliviment

My english desenvolviment (Not sure about your meaning here. =]) It's very slowly, but i don't won't (or will not) will give up. I hope soon I can whrite write a litlle little better than now.

 

My english (progress, development, evolution) is very slow, but I will not give up. I hope soon I can write a little better than now. 

Brazilians people

I am brazilian, and I stay am very sad when I see brazilian movie because this the movies show only bad language and slangs.(Not sure what you mean there.)  I want say one thing to everybody: The brazilian people are very polite and kind.We have problems, I know it, but Brazil is a beautiful and amazing country.  I am still learning English, and one day I want to know the Usa and Canada.

Busy Life.

I've been very busy recently.
I have 35 hours of lessons every week, and I go to language school to learn English every weekend.
This evening I went to downtown with my roommates and our professor, Shafik. Shafik is an Indian man who had  been living lived in London for a long time. He is smart, and funny, and kind. We had a good time.
I caught a cold a few days ago. The cold make is making me feel uncomfortable at present, but I can't go to bed until I finishi my English homework. I hope I will be better soon.

 

Very good!