Question

Consider a differential equation of the form y′=f(αt+βy+γ)y′=f(αt+βy+γ), where α,βα,β, and γγ are constants. Use the change of variable z=αt+βy+γz=αt+βy+γ to rewrite the differential equation as a separable equation of the form z′=g(z)z′=g(z).

Answer 1

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).