I was testing a certain solution for a differential equation and trying to determine whether it really is a solution or not; That's how it went:
$x=-y =>$
$(2\cdot - y -y)dx - (-y+y)dy = -3ydx = 0 $ . From here I have to infer whether $x=-y$ is indeed a solution to the differential equation or not. Nonetheless, I don't know how to infer it from the last expression.
Eventually the question boils down to whether or not $-3ydx = 0 $. Which is something I can't determine.
The "= 0" is confusing. "$2\cdot -y- y$" would be more easily written "-3y" and $(-y+ y)$ is 0. So apparently the equation is simply "-3y dx= 0". If x= -y then y= -x so -3y= 3x. No, -3y dx= 3xdx is NOT 0. In fact the general solution to -3ydx= 0, which is the same as dx= 0, is that x is a constant.