Understanding the Simplified Solution of a Differential Equation

[relinquished™]

I'm asked to find the general solution of the differential equation

$$x^2dx + y(x-1)dy = 0$$

I obtained a solution of

$$\frac{1}{2}x^2 + x + ln | x-1 | + \frac{1}{2}y^2 = C$$

The book, however, gives an answer of

$$(x+1)^2 + y^2 + 2ln |C(x-1)| = 0$$

I'm sure it's a simplified answer of my own answer. What I don't understand is how a term of +1 and +ln c appeared in the equation after transposing it in the equation. I know for a fact that you can transofrm c into (ln c) since they are constants, but the rest I don't get.

Thanks in advance.

 

[xepma]

Well, first of all you can multiply your equation with 2, and rewriting you constant to -(2Ln[C] +1) (which is ofcourse just another constant). After that the other answer more or less rolls out:

$$x^2 + 2x + 2ln | x-1 | + y^2 = -(2ln|C| + 1)$$
$$x^2 + 2x + 1 + y^2 + 2 ln | x-1 | + 2ln|C| = 0$$
$$(x+1)^2 + y^2+ 2 ln|C(x-1)| = 0$$

 

[relinquished™]

i see. I never knew you could add numbers to C. Thanx :)