Existence and Uniqueness of solutions

[STEMucator]

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 f_y 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 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 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.

 

[STEMucator]

Sorry for the double post, but I think I have something to add. What I was confused about is if I had to pick values for δ₁ and δ₂ as to make the solution continuous throughout the interval. Solving the equations he gave is actually quite difficult, so I'll give a simpler example to get my point across.

So for example, suppose I had y' = 1 + y² with the condition y(0) = 0 and I = ℝ.

Then f(t,y) = 1 + y² and f_y = 2y are defined and continuous everywhere for all (t,y) in I. Thus by our theorems, we can conclude that a solution exists in some open interval centered around 0 and the solution will be unique in some possibly smaller interval also centered at 0.

Solving the equation by separation yields y = tan(x). Notice that tan(x) has discontinuities at ±π/2, ±3π/2, ±5π/2, etc. So the solution is not continuous through the whole interval I.

So for y=tan(x) to be a solution, we have to restrict the domain of t to be in the open interval (-π/2, π/2).

So while we can pick δ as large as we like since f(t,y) is continuous, we would have to pick δ₁ = δ₂ = π/2 as to make our solution unique.

I'm having a bit of trouble relating this to the much more difficult equations given.