I am working with the below differential equation, where $k$ is a positive integer.
$\frac{d^2 y}{dx^2}+y(1+x^k)=0$
I need to show that the nontrivial solutions of this equation have infinitely many zeroes on the interval $(0,\infty)$ and also that the separation between adjacent zeroes goes to zero as $x$ tends to $\infty$.
I have read about Sturm-Liouville systems and finding the eigenvalues/eigenfunctions of such a system, but not sure how I am supposed to apply the theory in this case. Also, I have to show that solutions of the equation have infinitely many zeroes on an interval rather than the equation itself having zeroes, which is what I have seen in my notes. Is there a way to apply the Comparison Theorem to get the relevant result?
I will answer myself: employ the Sturm Comparison Test, which compares the zeroes of the solutions of two differential equations. The given equation is the 'hard' equation, but we can choose another simpler equation by setting the coefficient of $y$ to be $\omega^2 = x^k$. The situation with $\omega^2$ has well-known solutions from Sturm-Liouville theory. Since $x^k$ is less than $x^k +1$ on $(0,\infty)$, between any two adjacent zeroes of $y_1$ there is at least one zero of $y_2$ by the Sturm theorem. However, the 'easy' equation has a solution $y_1=\sin \omega x$ with an infinite number of zeroes at $x=n \pi/\omega$. It follows that the nontrivial solutions of the 'hard' equation have infinitely many zeroes in $(0,\infty)$.
To show that the separation tends to $0$ in the limit as $x$ tends to $\infty$, consider the interval where $y_2$ has at least one zero. The separation between zeroes is less than a particular quantity, so take the limit and show that this quantity goes to zero as $x$ goes to $\infty$.