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).

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).

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).

an

y = 6x -3y -12

y +3y +12 = 6x

=> . x = { 2y / 3 +2 } this is equation of line in terms of x

=> . if line meets the x-axis the co-ordinates of meeting point would be ( x , 0 )

=> . x = 2 ( 0 ) / 3 + 2

=> . x = 2

so, the point is ( 2 , 0 )

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