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.)

Try computing the derivative of each member. You have to use the Fundamental Theorem of Calculus:
$$-f'(x)=2f(x)$$
Now, divide by $f$:
$$-\dfrac{f'(x)}{f(x)}=2$$
Note that the first member is the derivative of $-\ln f(x)$. Can you continue?
