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Homework Statement
These questions were on my midterm a while ago. I want to understand this concept fully as I'm certain these will appear on my final tomorrow and I didn't do as well as I would've liked on these questions.
http://gyazo.com/205b0f7d720abbcc555a5abe64805b62
Homework Equations
Existence : Suppose f(t,y) is a continuous function defined in some region R, say :
R = { (x,y) | x0 - δ < x < x0 + δ, y0 - ε < y < y0 + ε }
containing the point (x0, y0). Then there exists δ1 ≤ δ so that the solution y = f(t) is defined for x0 - δ1 < x < x0 + δ1.
Uniqueness : Suppose f(t,y) and fy are continuous in a region R as above. Then there exists δ2 ≤ δ1 such that the solution y = f(t) whose existence is guaranteed from the theorem above is also a unique solution for x0 - δ2 < x < x0 + δ2.
The Attempt at a Solution
Okay, I'll start by discussing the first dot y' = 1 + y + y2cos(t), y(t0) = y0 on I = ℝ.
Suppose f(t,y) = 1 + y + y2cos(t), then fy = 1 + 2ycos(t). Notice both f and fy are continuous for all (t,y) in I. Thus by our theorems, we can conclude that a solution exists in some open interval centered around t0 and the solution will be unique in some possibly smaller interval also centered at t0.
This looks like a Riccati equation to me. I'm not sure if I should solve it, or continue my argument here.
Any pointers would be great.