# Homework Help: Differential equation

1. Jun 30, 2010

### thereddevils

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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?

2. Jun 30, 2010

### Staff: Mentor

Yes.
Above, e^c' is just a constant, so you can replace it by, say A.
That's a good idea.
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.