I'm studying DE's on my own to fill in gaps in my knowledge. I've come across a DE which, when I follow the procedure, results in a solution that doesn't satisfy the original DE. Here's the DE:

The DE is homogeneous, and either substitution (

or

) is, by symmetry, the same process. I choose

with

Then the DE becomes

For the first integral, let

, and then

or

and hence we get

This is our implicit solution. Differentiating with respect to

gives

This is not the original DE, nor do I see a method for getting the original DE. I conclude that either my solution is incorrect, or my differentiation is incorrect. Where's my mistake?

Thanks in advance!