HIERARCHAL INDUCTIVE PROCESS MODELING AND ANALYSIS ...

In addition to these 2 Lemmas, let us introduce a method for solving a first order
linear differential equation.
Proposition 1 Suppose u is a function that satisfies:
du
dt = αu + β, u(0) = u 0,
then,
u(t) = (u 0 + α β )eαt − α β .
Proof:
du
dt
= αu + β is a first order linear differential equation and can be solved
using the method of integrating factor.
(e −αt u) ′ = βe −αt
e −αt u =
∫ t
0
βe −αs ds
u = − β α + Ceαt
Using initial condition u(0) = u 0 we find the following solution:
u = (u 0 + β α )eαt − β α
Hence, proving proposition 1. □
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