I'm not quite positive where to even begin with this problem. Can somebody help me interpret what it is I need to do and/or how to find the correct solution?

By the Picard-Lindelöf existence and uniqueness theorem (also known as Picard's existence theorem or Cauchy–Lipschitz theorem), we can say that $x^\prime = x^{2}$, $x(0) = x_{0}$ has a unique solution on an interval $[-b/M , b/M]$ where $$ \sup_{x \epsilon B_{b} (x_0 )} \parallel f(x) \parallel $$. For $x_0 > 0$, show that $M = (x_0 + b)^2$ and then find a value of $b$ to maximize this interval.



The first thing you want to do is explicitly solve this ODE. I will write $y' = y^2$ and $y(0) = x_0$, using the variable $y$ for the sake of convention. The solution to this is

$$y(x) = \frac{1}{c - x}$$

when $x=0$, you have $1/c$, so set $c = 1/x_0$. You can see that this will not have a solution that exists for all time, since this function blows up at $x=c = 1/x_0$. Thus you will want to assert that the solution exists for $|x| < c$. Now try and rewrite this condition in the form you are asked.


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