Maximal Solution of an IVP I am able to do Q1 it's just the result is required in Q2.

I'm having issues framing the proof of Q2 and am struggling with its conclusion .
My attempt:
Let $J(x_0)$ be the union of all open sets  $J_i\ni0$ st. $x_{J_i}$ is a solution of 
$x'=f(x),x(0)=x_0  \quad (*)$,
so $J(x_0)$ takes the form $(\alpha(x_0),\beta(x_0))\ni0$
If i pick a $t$ st $t\in J_a \cap J_b$ where ,
$x_{j_a}:J_a \rightarrow W \quad $and $\quad x_{j_b}:J_b \rightarrow W $ are both solutions to $(*)$
By Q1:
$x_{j_a}:J_a \cap J_b\rightarrow R \subset W $ and
$x_{j_b}:J_a \cap J_b\rightarrow R \subset W $
where $R$ is the unique solution on the intersection $J_a \cap J_b$.
The union of all such intersections of $J_i \subset J(x_o)$ will  likewise have a unique solution and the solution will be defined on the entirety of $(\alpha(x_0),\beta(x_0))$. This solution is maximal because no $J_i$ is defined outside of this interval QED.
As you can tell the conclusion is not particularly detalied
 A: It is basically correct. You can take the union of all open intervals $J_i$, you don't really need to consider general "open sets" (the domain of a solution is assumed to be an interval in this kind of IVP). Union of open real intervals that have a point ($0$) in common, is an open real interval again. 
If $J(x_0)$ is the union of all such intervals and $t\in J(x_0)$, you can simply define $x(t)$ to be $x_a(t)$ where $x_a: J_a\to W$ is a solution of the IVP defined on some interval $J_a$. Part Q1 shows that this is well-defined: if $x_b: J_b\to W$ is another solution s.t. $t\in J_b\cap J_a$, then, by Q1, $x_a(t)=x_b(t)$. Which is basically what you have written. Further, the $x'=f(x)$ is satisfied in each $t$, because $t\in J_a$ for some $a$ and $x(t)=x_a(t)$ in some neighborhood of $t$. The initial condition is clearly satisfied as well.
To show maximality, if some solution were defined outside $J(x_0)$, then there would be a $\tilde{J}$ and a solution $\tilde{x}: \tilde{J}\to W$ with $\tilde{J}\nsubseteq J(x_0)$. But this is a contradiction, because $J(x_0)$ was defined to be the union of all domains of solutions, so $J(x_0)=\cup \{\tilde{J}, J_a, J_b,\ldots\}$ -- this clearly contains $\tilde{J}$.
