Separable equations: 1st order DE

[jenzao]

Homework Statement

solve the quation dy/dx = (4x - x^3) / (4 + y^3)
the first thing the book does is rewrite the equation as:

(4+y^3)dy = (4x-x^3)dx

and i understand that they are 1st separating it out... BUT shouldn't it be (1 / (4+y^3))dy?

How can they dissmiss the fact that the y terms are in the denominator?
On every problem, the fact that terms are under denominator gets ignored --why?

Homework Equations

thanks!

The Attempt at a Solution

 

[rock.freak667]

\frac{dy}{dx} = \frac{4x - x^3}{4 + y^3}

\times (4 + y^3)


(4 + y^3) \frac{dy}{dx} = 4x - x^3


Now separate the variables.

 

[tiny-tim]

How can they dissmiss the fact that the y terms are in the denominator?

Hi jenzao!

The y terms are in the denominator on the RHS,

but when you move them over to the LHS, they must go on top.

(and the dx on the bottom of the LHS must go on the top of the RHS for the same reason)

Technically, that's because if A/B = C/D, then AD = BC.