How can I get the (Volterra) operator from an equation of the type
I know that there is a general way of doing it, if you could point me at the proper book I'd be thankful!
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I understand that you want to rewrite the differential equation in terms of an integral (Volterra-type) operator. The resulting operator $T$ will be Hilbert-Schmidt, hence compact, hence $I-T$ is Fredholm.
Introducing $v=u'$, we get the system of 1st order equations $u'=v$, $v'=-u-xv$. Using the initial values $(u_0,v_0)$, we rewrite the IVP as a system $$u(t)=u_0+\int_0^t v(s)\,ds, \qquad v(t)=v_0+\int_0^t [-u(s)-xv(s)]\,ds$$ The desired operator $T$ takes the vector-valued function $(u,v)$ and produces $$t\mapsto \left(u_0+\int_0^t v(s)\,ds, v_0+\int_0^t [-u(s)-xv(s)]\,ds\right)$$