 After ignoring function "c", the first 2 equations give

and can be used as a third equation, which takes the form
The solution of is written in the form

where α0, ξ0 are positive constants, and τ0 is the upper limit of the era for the variable τ.It is convenient to introduce further a new variable (instead of τ)
Then
Equations and are transformed by introducing the variable χ = α − β