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.)

Differentiate the equation with respect to $t$ and use the Fundamental theorem of calculus: We obtain : 
$$ - \frac{ d f }{dt } = 2 f(t) $$
Next, write this equation as follows:
$$ \frac{df}{f} = - 2 dt \implies  \int \frac{df}{f} = - 2 \int dt \implies \ln f = -2t + C \implies f(t) = e^{-2t + C} $$
