if f'(x)=2f(x) and f(2)=1, then f(x)= ?

Answer 1

f'(x)=2f(x)
f'(x)/f(x) = 2

substitute u=f(x) and integrate both sides,
du/u = 2
ln(u)=2x + C
u=e^(2x+C)
f(x)=e^(2x+c)
f(2)=e^(2*2+C)=1 => C=-4

So f(x)=e^(2x-4)