Understanding Lipschitz ODEs

Question

Through other questions in Math Stack Exchange (like this one about Uniqueness), I was introduced to the Picard–Lindelöf theorem, which proof the Uniqueness of solutions of Lipschitz kind of ODEs, but this theorem is done through the Banach fixed-point theorem which I don´t understand, but it is used a method known as Picard Iteration where it is found a Taylor series which matches the series expansion of the solution.

Because of this Picard Iteration construction, I have a few questions I don´t find answers on Wikipedia or Google:

(1) Are the solutions to Lipschitz ODEs always analytical? Meaning here that can be represented through a Power Series.

(2) Is the maximum rate of change of the solutions to Lipschitz ODEs always bounded $\sup_t \|\dot{y}(t)\| < \infty\,\forall t$? If answer to question (1) is affirmative, it will follow since analytic functions are smooth - but I am not sure about the answer to question (1).

(3) Is the following equation a Lipschitz ODE?

In this another question I find an ODE that have a finite duration solution: $$ \frac{\dot{y}}{y} = -\frac{2\,t\,(1+|1-t^2|)}{(1-t^2)^2},\,\,y(0)=1\,\,\Rightarrow\,y(t)=\textstyle{\frac{(1-t^2+|1-t^2|)}{2}}e^{-\frac{t^2}{1-t^2}}$$ Since it achieve zero at an ending time and remains in zero forever after, it couldn´t be Lipschitz, Isn´t it? (for more details read this Finite Time Differential Equations (V. T. Haimo - 1985)). This is like a counterexample of question (1), if the solutions must be analytic, it can´t stand solutions similar the this one.

(4) Is the following another equation a Lipschitz ODE?

The following equation: $$\ddot{y}=\frac{\dot{y}}{t}-\frac{y}{t^2},\,\,y(0)=0\,\,\Rightarrow\,y(t) = \frac{t}{2}\log(t^2)$$ achieves an infinite maximum rate of change at $t=0$ without having a solution with a jump discontinuity, but its maximum rate of change happen in a point where its second derivative has a singularity, so I want to know if the Equation is Lipschitz or not (not the solution which I already know is not).

(5) Is the maximum rate of change of the solutions to Lipschitz ODEs always achieved at inflection points $\ddot{y}(t) = 0$?

I believe it is going to be true at least for every 2nd Order ODE which don´t have singularity in their former equation $F(\cdot)$ from $\ddot{y} = F(t,y,\dot{y})$ (as it does have the function of question (4)), this because for what I am asking here.

(6) Are the solutions to Lipschitz ODEs always Lipschitz continuous?

Answer 1

1. is wrong on the premise, it is only some demonstrative examples where the Picard iteration results in a sequence of polynomials. Take some ODE where the right side is non-polynomial and the sequence that you can compute manually or symbolically becomes very short.

2. That is trivially so, but that supremum need not exist. A maximum of continuous functions always exists on compact domains, so you get that property locally.

3. That ODE is only defined on the strip where $t\in(-1,1)$, your use of that example uses a continuation of the solution that is not the topic of the basic existence and uniqueness theory.

4. This ODE is not defined at $t=0$, so the theory only applies to $t\in(0,\infty)$. However, it has a regular singularity at $t=0$, moreover it is an Euler-Cauchy equation (or the homogeneous equation to it is), so depending on the exponents some solutions may have limits at $t=0$.

5. No. As usual that (the sufficiency of the second derivative condition) is only true if the maximum is inside the domain. Trivial counter-example: $y'=y$, $y''$ is not bounded.

6. Trivially true, at least in the local sense, as solutions to ordinary DE are by definition continuously differentiable, which implies locally Lipschitz.