How to solve the following integral equation in Mathematica?
eqn = y[x, z] == z + x*Integrate[y[t, v], {t, 0, 1}, {v, 0, 1}]
sol = DSolveValue[eqn, y[x, z], {x, z}]
What is the problem? Mathematica usually gives me answers like that while solving things that look difficult. I can solve simple problems. Difficult problems are why I need a software. Thank you in advance.
Mathematica can solve your equation, but it requires some understanding of mathematics and human intervention. This is always the case for nontrivial problems, isn't it?
Additive solution was demonstrated by @bbgodfrey. But there is another multiplicative one. Separate variables as follows
$$y(x,z)=g(x) h(z),$$ and set $$G=\int_0^1 g(x) dx.$$
Now your equation reads after integrating it $\int_0^1 dx\ldots$ $$ G h(z)=z+\frac{G}{2} \int_0^1\! h(v)\, dv. $$
This equation can be solved with MA
eqn = G* h[z] == z + 1/2* G*Integrate[h[v], {v, 0, 1}]
DSolveValue[eqn, h[z], z]
$$h(z)=\frac{2 z+1}{2 G}.$$
The full solution therefore reads
$$y(x,z)=\frac{(2 z+1)g(x)}{2 G},$$ where $g(x)$ is arbitrary function on the $[0,1]$ interval such that $G=\int_0^1 g(x) dx\neq0$.
x + z. $\endgroup$