y'=x-(y^2)sin(y) has unique solution?
My differential equations is rusted-to-the-point-of-useless, so here's a problem, what I did, and a question about it:
I'm staring at , with the condition that . I need to determine whether there is a unique solution here within some sufficiently small interval about zero. So, I put the y-relevant things on the left, the x-relevant things on the right, integrate both sides and end up with .
Now I'm not even sure I did that properly, but if I pretend I did then I get confused here because if I use the point (0, -2) to find C then graph the implicit function using this C, my graph looks like ellipse-ish forms radiating vertically from the origin--I'm unclear how to interpret this. Is this my unique solution?
If, on the other hand, I enter the original differential equation "as-is" with the specified condition into my graphing tool, a single line (however strangely shaped) appears before me, suggesting more directly (to my inept intuition, at any rate) an affirmative "Yes, there IS a unique solution, and you're staring at it, buddy".
Methinks a wholly analytic solution would show me the link I'm missing between the two, but I'm unsure how that goes. Maybe I've just been up too long to sort it out solo, but any clarification would be greatly appreciated.
Thanks in advance.