Solving and Elasticity Problem: Differential Equation

Homework Statement

Show that the general solutions to the equations:
$$(\frac{d^2}{dr^2}+\frac{1}{r})(\frac{d^2f_1}{dr^2} + \frac{1}{r}\frac{df_1}{dr}) = 0$$

$$(\frac{d^2}{dr^2}+\frac{1}{r}\frac{d}{dr} -\frac{4}{r^2})(\frac{d^2f_2}{dr^2}+\frac{1}{r}\frac{df_2}{dr}-\frac{4f_2}{r^2})=0$$

are:

$$f_1=c_1r^2\ln r + c_2r^2 + c_3\ln r + c_4$$
$$f_2=c_5r^2+c_6r^4+\frac{c_7}{r^2}+c_8$$

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The Attempt at a Solution

I'm not familiar with this kind of differential equation. I've got homogeneous linear higher order differential equations down (thanks to http://tutorial.math.lamar.edu/Classes/DE/HOHomogeneousDE.aspx ), but I'm not sure how to approach this one. A lead into what this is called or some characteristic equation and the general solution would be nice.

Cheers,
Justin