I can't get my head around why the kernel in the Volterra integral equation can't be separable. $$u(x) = f(x) + \int_a^x K(x,s)u(s)ds, x \in [a,b]$$ A separable kernel $K(x,s)$ is the one that can be rewritten in the form of $$K(x,s ) = \sum_{i=1}^NA_i(x)B_i(s ).$$ It's supposed to be a "text book" problem, and the hint is "$K(x,s)=0$ when $s>x$", which really doesn't help much at all. I tried to somehow use the fact that it can be rewritten as the Fredholm integral equation in a triangle area: $\hat{K}(x,s)=\begin{cases} K(x,s) & a\leq s\leq x\\ 0 & x\leq s\leq b \end{cases}$
but so far no luck.
Edit: It seems that for the condition "$K(x,s)=0$ when $s>x$" to hold, if we assume separability, then we can't effectively make it zero. Either $A_i(x)$ will depend on s ($A_i(x)=0$ when $x<s$) or $B_i(s)$ will depend on x ($B_i(s)=0$ when $s>x$). This contradicts separability. Thoughts?