Solving the ODE 4(dy/dx)=4-y^2 using separation of variables method

[thereddevils]

Homework Statement

Solve the ODE, 4(dy/dx)=4-y^2

Homework Equations

The Attempt at a Solution

separating the variables,

dy/(4-y^2)=dx/4

then integrating both sides

(1/4) ln((2-y)/(2+y))=x/4+c

Multiply by 4, so now the constant is different, so must i use a different variable for the constant?

ie ln((2-y)/(2+y))=x+c'

(2-y)/(2+y)=(e^x)(e^c')

y=(2-2(e^x)(e^c'))/((e^x)(e^c')+1)

then here, 2(e^c') is another constant, do i have to use another variable to represent it?

And also what's the definition of arbitrary constant?

 

[Mark44]

Homework Statement

Solve the ODE, 4(dy/dx)=4-y^2

Homework Equations

The Attempt at a Solution

separating the variables,

dy/(4-y^2)=dx/4

then integrating both sides

(1/4) ln((2-y)/(2+y))=x/4+c

Multiply by 4, so now the constant is different, so must i use a different variable for the constant?

Yes.

ie ln((2-y)/(2+y))=x+c'

(2-y)/(2+y)=(e^x)(e^c')

Above, e^c' is just a constant, so you can replace it by, say A.

y=(2-2(e^x)(e^c'))/((e^x)(e^c')+1)

then here, 2(e^c') is another constant, do i have to use another variable to represent it?

That's a good idea.

And also what's the definition of arbitrary constant?

When you evaluate an indefinite integral such as this --
$$\int x~dx = \frac{1}{2}x^2 + C$$
-- the constant C can be any number, so can't be determined, hence is arbitrary.