Hello
problem:
Solve :
It must be solved using inspection
which is , ... etc
But this equation is bothering me
I tried as much I can, and it seems impossible!
any help??
Well, aside from the trivial solution that I gave you before, nothing is obvious. It's not linear in either x or y, it's not exact, homogeneous, or separable. And I don't see any obvious substitution. All of this means that, unless you can manipulate the equation to the point where you can do a direct integration, you may not be able to find a general solution at all. I certainly see no possibility for a solution by inspection, except for what I have already given.
I tried a quotient rule with a function of two variables thus: make
Carrying out the differentiation and comparing to the original DE yields the two equations
and
Unfortunately, it's impossible to make those two equations work, as far as I can tell. You're close, actually, very close. I get
But plugging this into the second equation above yields
which is off by a minus sign and a fraction of 1/2.
This approach might be made to work, but I'm not seeing how.