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dN/dt = k - λN

All my textbook assumes that k is a constant vale.

http://jol.liljenzin.se/KAPITEL/CH15NY3.PDF

for the equation to be valid,

(iii) the flux is not decreased ... . ..

Corrections must then be introduced.

Please advise me that is there any reference that doing this correction.

What form of the equation will be, is there analytic solution ?