There are two questions wrapped up in this post: "What's wrong with my answer?", and "Explain downvoting behavior.". AlexanderGruber addresss the first question, so I'm going to respond to the second.
At least let me know is anything wrong in the solution so that I can correct it. ... I don't mind even the post gets deleted but there should be a valid reason for that. I'm extremely furious to this behavior of MSE users.
Based on my experience here (4 years), you can take it as an axiom that some of your posts (both questions and answers) are going to get down votes without an explanation. It will happen. I figure people either find a way to live with it, or they leave, or they stick around being furious (which I think isn't good for their mental health, or for the site).
One thing that can be useful is to realize that math.se isn't a classroom where one is working to receive a grade (which, let's face it, is how most of us spent our first 10-15 years doing math). Here we're all contributing to a body of work that is supposed to be useful to others. In the past I've submitted answers that I thought were a very cool way of looking at a problem, and gotten zero votes or down votes. Oh well, I enjoyed thinking about the question and coming up with the answer. Other people thinking it's not useful for the site doesn't take that away from me.
There is another question you raised: "How can I get a critique of my answer if it's getting down voted?". Remembering that this isn't guaranteed to anyone, we realize that we're asking others to spend some extra time and effort doing something for us. So sometimes adding a comment "Curious about the downvotes?", or "Are the downvotes because of an error in the answer or something else?" can prompt a response. For myself, I might not bother to explain a downvote - it might feel like wasted effort - but if someone specifically asks for an explanation I'd think "oh, it is worth my effort adding an explanation here".
Another possibility is chat rooms. Stealing @ArcticChar's comment: Constructive Feedback is a chat room where people can go with a "what's wrong with my answer" query.