The question is:
Examine a differential equation expressed as
y′ = f(αt + βy + γ),
where α, β, and γ are constants. Employ the variable change
z = αt + βy + γ to reformulate the differential equation as a separable equation of the type z′ = g(z).
Answer:
The equation
y′ = f(αt + βy + γ)
can be rephrased as
dy/dt = f(αt + βy + γ).
Our goal is to rewrite this differential equation in the form
z' = g(z), that is dz/dt = g(z).
First, be aware that
dz/dt = (dz/dy) * (dy/dt)...................(1)
Utilizing the substitution
z = αt + βy + γ
as specified,
dz/dy = β..........................................(2)
dy/dt = f(αt + βy + γ) = f(z)............(3)
From equations (2) and (3),
dz/dt = β.f(z) = g(z)
Thus,
z' = g(z)
Where g(z) = βf(z).
