If f(x) = cos x –∫_0^x(x-t)f(t)dt, then f’’ (x) + f(x) equals

$f(x) = \cos x –\int_0^x(x-t)\,f(t)\,dt$

$f(x) = \cos x –x\int_0^xf(t)\,dt+\int_0^xt\,f(t)dt$

$∴ f’(x) = –\sin x –\left\{xf(x)+\int_0^xf(t)\,dt\right\}+ x f (x)$

$= –\sin x –\int_0^xf(t)\,dt$

$∴f’’(x) = –\cos x – f(x) ⇒ f’’(x) + f(x) = –\cos x$