My PhD advisor wants me to work on 2 different problems during my PhD. One is the problem which she had suggested and another is collaborative work with some company. Both the problems are disjoint. My PhD advisor has given me the option, that if I want to work on one problem under her--then it should be the problem on which she is doing collaborative research and I should leave the problem which she has suggested. However, I dont want to give up my present problem (the one which she suggested) as that sounds interesting to me and I have already worked for 1 year on it. So, when I told her that I dont find the collaborative work interesting, then she suggested that I work on both the problems. However, I am finding hard to work on 2 problems. Its like a dual PhD for me. I can not say anything to my advisor as she says I have to work on the collaborative work as my stipend comes from it and else I should leave PhD. My advisor also does not advise me when I propose ideas about my previous research. Please suggest what should I do?
The most important line in your post seems to be one that you didn't give much weight to: "my stipend comes from [the collaborative project]". From my experience, this is a common situation, where the money for your stipend comes from a project you're not interested in pursuing.
Unfortunately, the only real alternative for you is to write your own grant proposal, which is a significant investment of time for both you and your advisor, and carries the not-so-minor risk that the grant won't be funded, which leaves you exactly where you started, with the added problem that there hasn't been much progress on the existing grant.
The second alternative is to leave the advisor and try to find someone else, but I would recommend against that for two reasons. Firstly, it's hard to find someone else with funding on a project you like. Secondly, the problem you're having now—the funded project is less interesting than the unfunded project—is one you'll likely face again in the future. Your advisor's approach of pursuing both projects is (again, from my experience) a typical way of dealing with this, and it will likely benefit you to have experience with this approach during your PhD years.
I recognize that it means extra work for you, and I don't really have much advice on that front. Try not to think of it as extra work, but rather as extra publications in different research areas. That helps minimize the sting.
It depends on how committed you are to the research project. Your Ph.D. thesis can strongly influence your future course in academia and research. That being said, plenty of Ph.D. recipients have gone on to do research in fields quite distinct from their thesis work.
If your funding is tied to one of the projects, it would certainly be easier to find the time to complete your thesis by working on that project alone. However, you should only make this shift if the funded project is also interesting to you. There few things as difficult as finishing a thesis on a topic in which you are not interested.
Ultimately, what matters most about your Ph.D. thesis is that you finish it. If you think that finding time to work on your thesis will be most limiting, then you should choose the funded topic. If you think that staying interested in your research is the biggest hurdle, then you should choose the topic you are interested in, even if it takes you longer to graduate on account of your other obligations.
There is nothing inherently wrong in an advisor asking you to work on two projects — depending on the subfield you're in, this could even be common practice. The two common approaches are:
I honestly can find no disadvantage with the second route except for the fact that it can be a bit overwhelming for a graduate student, especially if their advisor doesn't realize that the work load is heavy. I tend to look at it as training for the future — as a postdoc/scientist/faculty, you'll most likely be juggling several different projects of your own, in addition to collaborating, teaching, advising, etc. So this is a good way to start learning efficient time management.
In short, I don't think your situation is alarming and you should take this as an opportunity to learn some soft skills (time management, recognizing a hot potato, identifying paths that lead somewhere, etc.). Now, if you're really uninterested in the project, then that's a different issue — you probably have to look for a different advisor.