A function equal to its integration? It is asked that I find a function such that $$10-f(x)=2\int_0^xf(t)dt.$$ I tried giving a new function F(x) such that ${dF(x)\over dx}=f(x)$, but all I got was a new equation $$F(x)=10x-2\int_0^xF(t)dt.$$ So how do we find such function. Thanks in advance! (I am new to differential equations, so I do not know much about the topic yet.)

You have the right idea: Differentiating both sides gives
$$-f'(x) = 2f(x),$$
which has general solution $$f(x) = C e^{-2x}.$$ Evaluating both sides at a convenient value of $x$ (say, $x = 0$) determines $C$ uniquely.
