Verifying My Solution to an EDP: Agree or Disagree?

[Carmen_8]

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:
$$u=\sqrt{2\eta F(\xi )+2G(\xi )}$$

 

[DuckAmuck]

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?