Not that Snake again!

$\displaystyle f(x)=x+x\int_{0}^{1}y^2f(y)dy+x^2\int_{0}^{1}yf(y)dy= x+xc_{1}+x^2c_{2}$

where, $\displaystyle c_{1}=\int_{0}^{1}y^2f(y)dy=\int_{0}^{1}y^2(y+yc_{1}+y^2c_{2})dy=\frac{1}{4}+\frac{c_{1}}{4}+\frac{c_{2}}{5}$

and, $\displaystyle c_{2}=\int_{0}^{1}yf(y)dy=\int_{0}^{1}y(y+yc_{1}+y^2c_{2})dy=\frac{1}{3}+\frac{c_{1}}{3}+\frac{c_{2}}{4}$

Solve to get: $c_{1}=\frac{61}{119}$ and $c_{2}=\frac{80}{119}$

$\Rightarrow f(1)=1+\frac{61}{119}+\frac{80}{119} \approx 2.185$