The question is:
Consider a differential equation of the form
y′ = f(αt + βy + γ),
where α,β, and γ are constants. Use the change of variable
z = αt + βy + γz
to rewrite the differential equation as a separable equation of the form z′ = g(z).
Answer:
The equation
y′ = f(αt + βy + γ)
can be written as
dy/dt = f(αt + βy + γ).
We want to rewrite this differential equation is the form
z' = g(z), that is dz/dt = g(z).
First, note that
dz/dt = (dz/dy).(dy/dt)...................(1)
Using the substitution
z = αt + βy + γ
as required,
dz/dy = β ..........................................(2)
dy/dt = f(αt + βy + γ) = f(z) ............(3)
From (2) and (3)
dz/dt = β.f(z) = g(z)
So,
z' = g(z)
Where g(z) = βf(z).
