Given $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , what is $f(0) $?

Question

Let $f:\mathbb R \to \mathbb R$ be such that $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , then how do we find $f(0) $ ?

Answer 1

$f(x)=x+\int_{0}^1 t(t+x)f(t) dt=c_1x+c_2 $ where $c_1=1+\int_{0}^1 tf(t) dt $ and $c_2=\int_{0}^1 t^2f(t) dt $.

Put $f$ in the integral and evaluate, you can get a linear system in terms of $c_1$ and $c_2$. Solve it you get an expression of $f$.