Question: integration problem

I am trying to integrate a nonlinear fourth order PDE.





To get a third order nonlinear PDE.

I am unsure exactly what happens to this equation when it is integrated with respect to η.

Here is what I think should happen.

Fηηηη(η,t) will integrate to Fηηη(η,t)

ηαFηηη(η,t) will integrate to ηαFηη(η,t)-αFη(η,t) by parts

-aFηηt(η,t)/ν will integrate to -aFηt(η,t)/ν

3αFηη(η,t) will integrate to 3αFη(η,t)

F(η,t)Fηηη(η,t)-Fη(η,t)Fηη(η,t) can be integrated by parts to get

F(η,t)Fηη(η,t)-Fη(η,t) so long as we realise that Fη(η,t)Fηη(η,t) can be written as [½Fη(η,t)]η

So I should obtain

[Fηηη(η,t)+ ηαFηη(η,t)-αFη(η,t) -aFηt(η,t)/ν+3αFη(η,t)+ F(η,t)Fηη(η,t)-Fη(η,t)]η=0

Is this correct? And how can I use Maple to check if the integration is correct?

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