Solution of an integral equation $\phi(x)+\int^1_0 xt(x+t)\phi(t)\,dt=x $ , $0 \le x \le 1 $ Solve the following integral equation:  
$\phi(x)+\displaystyle \int^1_0 xt(x+t)\phi(t)\,dt=x $ , $0 \le x \le 1 $

I need to solve the integral equation above. Can anyone help me please?

Hint: $$\phi(x)+x^2 \displaystyle \int_0^1t\phi(t)dt+x\int_0^1t^2\phi(t)=x$$ 
i.e. $$\tag{1}\phi(x)=c_1x^2+c_2x $$
Where $c_1 = - \displaystyle \int_0^1t\phi(t)dt \ \ $ and $\ \ c_2=1- \displaystyle \int_0^1t^2\phi(t)dt$
Put $\phi(t)$ from $(1)$ in these integrals and calculate  $c_1 \ , \ c_2$ from two equations you get 
(two-equations  two variables).
