We differentiate both sides of (*) and obtain

f(x) = 3f¢(x).

Thus if there is a solution of (*) it must be of the form

f(x) = Aex/3,

for some constant A. We check to see whether or not this is a solution. Thus with f(x)=Ae^x/3 we have

ó õ x 0  Aet/3 dt = é ë 3Aet/3 ù û x 0  = 3Aex/3-3A.

Thus f(x)=Ae^x/3 is a solution when (and only when) A=-k/3. Thus the unique solution is

f(x) = -k 3 ex/3.