Existence and Uniqueness Theorems: When to use Picard-Lindelöf, Lipschitz, etc.

Question

I am currently reviewing some basic ordinary and partial differential equations for an upcoming oral exam and I am stuck at existence and uniqueness theorems.

As far as I understand, one would like to know two things about a (relatively) complicated ODE before trying to find a solution.

1. Does the ODE even have a solution?
2. If it has a solution, is it unique?

Consider the IVP:

$$\frac{dy}{dt}=f(x,y); \space \space \space y(a)=b$$

Question1: Suppose $f$ satisfies the Lipschitz condition. What does that mean and what does that tell me about the solution of my ODE?

Question 2: When and how does the Picard-Lindelöf theorem come in? Does Picard-Lindelöf only tell me about the uniqueness? Do I need Lipschitz continuity in order to even use Picard-Lindelöf?

If someone asked me to specify in which domain $D \subset \Bbb R^2$ the differential equation $(x^2+y^2)y'=y^2$ has a unique solution, how would I approach this problem (or any other similar problem)?

Answer 1

Consider a first order linear differential equation $y'+Py=Q,\,y(a)=y_0$ .Here continuity of $P$ and $Q$ ensure that the the ODE has an unique solution.But in case of non-linear initial value problem i.e $y'=f(x,y),\,y(x_0)=y_0$,continuity of $f$ does not ensure the unique solution.So we need to address the following question:

(1)Under what condition on $f$ the problem $y'=f(x,y),\,y(x_0)=y_0$ has a solution$?$

(2)If solution exists,whether it is unique or not$?$

$\to$First question is answered by the Peano existence theorem which states that "let $f$ be a continuous function in an interval $I$ containing the points $(x_0,y_0)$,then the the problem $y'=f(x,y),\,y(x_0)=y_0$ has a solution".

$\to$Second question is answered by Picard's uniqueness theorem which states that "let $f$ and $\frac{\delta f}{\delta y}$ are continuous in aregion R containing the initial points $(x_0,y_0)$ then the the problem $y'=f(x,y),\,y(x_0)=y_0$ has an unique solution"

Picard method for interval of definition:let $f$ and $\frac{\delta f}{\delta y}$ are continuous in a closed rectangle $$R=\{(x,y):|x-x_0|\leq a,|y-y_0|\leq b\}$$.Then the IVP $y'=f(x,y),\,y(x_0)=y_0$ has an unique solution in the interval $|x-x_0|\leq h=min{(a,\frac{b}{l})}$ where $l=MAX_{(x,y)\in R}|f(x,y)|$

Hope this will help you!!!