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I am trying to find the exact solution of:

u'''' + 5u =f on (0,1) f is a constant >0

where

u(0)=u'(0) =0

u''(1) =k constant >0

u'''(1)=0

I assumed I should look at solving the homogeneous equation and got the following

uh = A*exp(Zx)*sin(Zx) + B*exp(ZX)*cos(Zx) + C*exp(-Zx)*sin(Zx) +D*exp(-Zx)*cos(Zx)

Where Z is (sqrt(2)/2)*sqrt(sqrt(5)) and A,B,C,D are constants.

My questions are:

1) Am I on the right track

2) How would I go about solving for the particular solution

3) Does the exact solution actually exist?

Thanks