Independent functions as solutions

[Sathish678]

TL;DR Summary

I found three independent functions as solutions for this equation
d/dr(r^2dR/dr) = 6R (cauchy equation)
r^2 , r^(-3) , (1/7)r^6.

I found three independent functions as solutions for this equation
d/dr(r^2dR/dr) = 6R (cauchy equation)
r^2 , r^(-3) , (1/7)r^6.
But i read that a second order linear differential eqn has only two independent solutions.
Why this happened?

 

[pasmith]

$\frac17 r^6$ is not a solution. If it were, then as the equation is linear and homogenous $r^6$ must itself be a solution. It is not: $$\frac{d}{dr} \left(r^2 \frac{d}{dr}(r^6) \right) = \frac{d}{dr} \left( 6 r^7 \right) = 42r^6 \neq 6r^6.$$

 

[HallsofIvy]

In a bit mote detail, if $R= \frac{1}{7}r^6$ then $\frac{dR}{dr}= \frac{6}{7}r^5$. So $r^2\frac{dR}{dr}= \frac{6}{7}r^7$ and then $\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)= 6r^6$. That is NOT equal to 6R because it is missing the "$\frac{1}{7}$".