<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] $$f(x) = x + \int_{0}^{1} (xy^2 + x^2y) f(y) dy$$
$$f'(x) = 1 + \int_{0}^{1} (y^2 + 2xy) f(y) dy$$
$$f''(x) = \int_{0}^{1} 2y f(y) dy=\text{constant}$$
$$f(x)=ax^2+bx+c$$
$$ax^2+bx+c=x+ \int_{0}^{1} (xy^2 + x^2y) (ay^2+by+c) dy$$
After calculus of the integral and simplification :
$$(-\frac13 a+\frac13 b+\frac12 c)x^2+(1+\frac15 a-\frac34 b+\frac13 c)x-c=0$$
$$\begin{cases}
-\frac13 a+\frac13 b+\frac12 c=0 \\
1+\frac15 a-\frac34 b+\frac13 c=0 \\
c=0
\end{cases} \quad\implies\quad 
\begin{cases}
a=\frac{80}{119} \\
b=\frac{180}{119}
\end{cases}$$
$$\boxed{f(x)=\frac{80}{119}x^2+\frac{180}{119}x}$$
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