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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) | x

_{0}- δ < x < x

_{0}+ δ, y

_{0}- ε < y < y

_{0}+ ε }

containing the point (x

_{0}, y

_{0}). Then there exists δ

_{1}≤ δ so that the solution y = f(t) is defined for x

_{0}- δ

_{1}< x < x

_{0}+ δ

_{1}.

**Uniqueness :**Suppose f(t,y) and f

_{y}are continuous in a region R as above. Then there exists δ

_{2}≤ δ

_{1}such that the solution y = f(t) whose existence is guaranteed from the theorem above is also a unique solution for x

_{0}- δ

_{2}< x < x

_{0}+ δ

_{2}.

## The Attempt at a Solution

Okay, I'll start by discussing the first dot y' = 1 + y + y

^{2}cos(t), y(t

_{0}) = y

_{0}on I = ℝ.

Suppose f(t,y) = 1 + y + y

^{2}cos(t), then f

_{y}= 1 + 2ycos(t). Notice both f and f

_{y}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 t

_{0}and the solution will be unique in some possibly smaller interval also centered at t

_{0}.

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.