calculus

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

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  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)

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