IE4

Find lambda: $y(x)=\lambda\int_0^1 x t y(t) dt\,$ if $y(x)=x\,$

If $y(x) = x\,$ , then

$y(x)=\lambda\int_0^1 xty(t)\ dt\,$	$\Longrightarrow x = \lambda\int_0^1 xtt\ dt\,$
	$\Longrightarrow x = x\lambda\int_0^1 t^2\ dt\,$
	$\Longrightarrow 1 = \lambda\left(\frac{1}{3}t^3\left.\right\|_0^1\right)\,$
	$\Longrightarrow 1 = \frac{\lambda}{3}\,$

Therefore, $\lambda = 3\,$ .