A Verifying My Solution to an EDP: Agree or Disagree?

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I have this EDP, uu_{\eta \eta } +u_{\eta }^2=0
and my solution for it, but I don't know if it's correct. Could you tell me if you do or do not agree with it?
Here it's my solution:
1)We can write the EDP as:
(uu_{\eta})_{\eta}=0
2)Integrating:
d(uu_{\eta})=0
uu_{\eta}=F(\xi )
where \xi is another variable.
3)If we separate variables and we integrate again, we obtain:
u=\sqrt{2\eta F(\xi )+2G(\xi )}
 
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Seems correct to me. Although I'm not sure why you say F(x) G(x), why are they dependent on the same variable? Why not just say F and G are constants?
 
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