<s>[INST] $f(x) = x + \int_{0}^{1} (xy^2 + x^2y) f(y)dy$. Find $f(x)$ $f(x) = x + \int_{0}^{1} (xy^2 + x^2y) f(y) dy$. Find $f(x)$ 
I've tried taking $\int_{0}^{1} (xy^2 + x^2y) f(y) dy$ to be $k(x)$ since it comes out to be a function of $x$. That transforms our equation to $f(x) = x + k(x)$.
$f(y) = y + k(y) \implies (xy^2 + x^2y)f(y) = (xy^2 + x^2y)y + (xy^2 + x^2y)k(y)$
Now I tried integrating on both sides against $dy$ from $0$ to $1$ in an attempt to find $k(x)$ 
$\int_{0}^{1} (xy^2 + x^2y)f(y).dy = k(x) = \int_{0}^{1} ((xy^2 + x^2y)y + (xy^2 + x^2y)k(y)).dy$
But I got stuck trying to integrate the right hand side. Any solutions or ideas are appreciated. 
 [/INST] Let $A=\int_0^{1} y^{2}f(y)dy$ and $B=\int_0^{1} yf(y)dy$. Then $f(x)=x+xA+x^{2}B$. Multiply by $x$ and integrate to get $B=\frac 1 3 +\frac A 3 +\frac  1 4 B$. Similarly multiply by $x^{2}$ and integrate to get $A=\frac 1 4 +\frac A 4 +\frac  1 5 B$. Solve these two equations for $A$ and $B$. 
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