There are two integral equations of $$\alpha(x,y)=\int_x^y K(x,y,t)U(t)dt$$ $$\beta(x,y)=\int_x^y K(x,y,t)U(t)tdt$$
where $K(x,y,t)$ is a non-negative known function which $t$ lives in $[y,x]$; $U(t)$ is an unknown function in $(-∞,∞)$ that is differentiable; $U(0)=0$; $α(x,y)$ and $β(x,y)$ are known; $x$ and $y$ are independent. What condition must be satisfied by $K(x,y,t)$ in order to find $U(t)$, or $U(α)$ and $U(β)$? Analytically or numerically?
Otherwise, if only a list of data points $(x,y,α,β)$ is given, can $U(t)$ still be found?
I am not a math student and have little knowledge of math. I tried to read a book while I noticed this may relate to the field of Integral Equations but it is too difficult for me to understand a bit. I would like to know if there is a situation these equations are solvable and what particular method can I use. In fact, the second case is the one in real life.
If for certain case (eg. polynomials), $U(t)$ can be solved, I am thinking maybe I can use something like the Taylors Series to approximate the $K(x,y,t)$.