[tex]

y'''' + f(x)y'' = 0

[/tex]

In this case, you have to assume that f(x) is too complicated to be written down in closed-form. I don't need a closed form solution---an integral equation will do. I can take Fourier Transforms, then get the equation

[tex]

(ik)^4\widehat{y} + (ik)^2 \int_{-\infty}^\infty \widehat{f}(s) \widehat{y}(k-s) \ ds = 0

[/tex]

Unfortunately, there doesn't seem a way for me to isolate [tex]\widehat{y}[/tex]. Is there a standard technique for dealing with these things?

I know, for example, that the Airy equation, y'' = xy has no closed-form solution, but there is a way to put it into integral form. The trick, however, is that you need to know how to take the Fourier transform of xy (which you can).