Can some one tell me if i am correct or completely off

with initial condition

here and are constants.

by substitution

therefore

using condition

the solution is

which can be factored as

is this correct?

this is in reference to the paper in Malaria Journal Smith et al 2007 'Standardising estimates of the Plasmodium Falciparum parasire rate' this equation is listed on page 3 and i cant recapture their solution

their solution is

I must be doing something stupid