Are these two general solutions or one general/two particular solutions?

[s3a]

Homework Statement

The problem and its answers are attached.

I prefer the first one as (same thing but isolated for y):

y_1(x) = [k_1 * x + x * ln(x)]/[k2 - ln(x)]

Homework Equations

First order homogeneous differential equation.
y = xv, dy/dx = v + x dv/dx

The Attempt at a Solution

I solved this successfully. Given that at some point in the solution of the problem for solving y_1(x), I get a denominator with (x+y), it is my "mathematical duty" to check if x + y = 0 => y = -x is a solution to the differential equation and it is. Because of the nature in which I got this y = -x solution, I am confused as to whether I treat this as one of the two alternatives for a general solution or both together span the one and only general solution.

Any input would be greatly appreciated!
Thanks in advance!

 

[Dick]

Your equation isn't linear, is it? It has a y^2 in it. That would mean a linear combination of solutions isn't necessarily a solution.