Consider this as a known fact:
For an ordinary differential equation expressed in the normal form $y''+q(x)y=0$, if $\forall x, x>0:q(x)>0$ and $\int_{1}^{\infty}q(x)dx\rightarrow\infty$, then $y(x)$ has infinitely many roots in the positive region of $x-$axis.
Now the equation $y''+\left(\frac{k}{x^2}\right)y=0$ is given and I have to show that for $k>\frac{1}{4}$ the $y(x)$ has infinitely many roots while the number of roots of $y(x)$ for $k\le\frac{1}{4}$ is finite. The only tool I know that helps me to prove such a thing is the above theorem but when you calculate the integral
$$\int_{1}^{\infty} \frac{k}{x^2}dx=-\frac{k}{x}\Bigl|_{1}^{\infty}=k$$
It is always bounded. Do you have any idea what is going on here?
Thanks in advance.