If I read this correctly, you are going to need a first order linear differential equation to model weight as a function of time. We may express this as the initial value problem:

where

where:

is weight as a function of time .

is daily caloric intake (assumed constant).

is the caloric expenditure as a function of weight (for simplicity, height and age are held constant).

We have:

where the constants are:

Letting:

we have:

Multiplying through by we have:

Using the initial conditions, we find:

and so: