solve the equation
$$ f(x) + \int_0^1 (xy+x^2y^2) f(y) dy = g(x) $$
and write in the form of
$$ \sum a_jx^{j-1} $$
I have tried integration by parts but it doesn't seem to work because of f(y).
Do you need to know what g(x) is to solve this equation?
Any assistance on the method will be much appreciated.
$\int_0^1 (x y^2 + x^2 y^2) f(y)\; dy$ must be of the form $s x + t x^2$ for some constants $s$ and $t$. Thus $f(x) = g(x) - s x - t x^2$. Now plug that in to the equation and solve for $s$ and $t$ in terms of $\int_0^1 y g(y)\; dy$ and $\int_0^1 y^2 g(y)\; dy$.