I have learned Peano existence theorem and Uniqueness of solutions to IVPs, but I don't understand what does that mean by $y'=f(x,y)$, $y(x_0)=y_0$ has a solution in $(x_0-d,x_0+d)$ for some $d>0$.
- What is this thm trying to show?
- How to understand its uniqueness?
I have a hw question following:
Prove that if $f$ is continuous on $\Bbb R× \Bbb C$ and locally Lipschitz in the second argument, and if $x_0 \in \Bbb R$, $y_0 \in \Bbb C$, then there exists an interval $(a; b)$ containing $x_0$ such that the following holds.
- A solution to IVP $y' = f(x; y)$ and$ y(x_0) = y_0$ exists on $(a; b)$, and
- if a solution $\tilde y$ to the IVP exists on some open interval $I$ containing $x_0$, then $I\subset(a; b)$ and $\tilde y = y$ on $I$. That is, (a; b) is the largest interval of existence and the solution is unique on it.
(Hint: Show that if $F$ is the family of all couples (open interval containing $x_0$, solution on it), then any two of these solutions coincide on the intersection of their intervals of definition. Conclude that then a solution can be defined on the union of these intervals (show it is an open interval) such that it coincides with each solution in F where the latter is defined.)
I just don't understand how is the hint related to the thm.
I know I have learned this part poorly, I really appreciate if you can let me understand the thm.