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!