I am confused with a problem I encountered at hand, not on how to work on it but rather understanding the problem itself:
Let $A(x;\omega)$ be a random field taking values in $[a,b]$ where $a,b < \infty$ with $\omega \in \Omega$.
Consider the differential equation $\frac{d}{dx} (A(x;\omega)y'(x))=0$ subject to $y(0)=0,\,y(1)=1, \,0<x<1.$
For a fixed $\omega$, I know that $A(x;\omega)$ is a real valued function. I aim to construct several samples of $A(x;\omega)$ for say 10000 different $\omega$'s and use Monte Carlo to find the solution $y(x).$ But how do I construct multiple instances of $A(x;\omega)$ which takes values on $[a,b]$?
Of course one such instance is $A(x) = a + (b-a)U$ where $U$ is uniformly distributed on $[0,1]$ but in this case, won't the above differential equation be very trivial to solve since $A(x;\omega)$ is constant for each $\omega$?
Hopefully someone can shed light on this...
To simulate the random field $A$, one needs more information than the fact that $A(x)\in[a,b]$ almost surely, for every $x\in(0,1)$. A slightly less degenerate version of your proposal is to define $A(x)=a+(b-a)U^x$ for every $x\in(0,1)$, but, once again, we cannot know unless you explain what $A$ should be.
Note that one can directly solve the differential equation as $$y(x,\omega)=\frac{Z(x,\omega)}{Z(1,\omega)},\qquad Z(x,\omega)=\int_0^x\frac{\mathrm du}{A(u,\omega)}.$$